Over a span of five days from June 6 to 10, we will be running a series of one-day mini conferences in London, Zurich, Berlin, Eindhoven, and Warsaw.
By decreasing everyone’s travel time, reducing the time commitment to one day, and slashing the cost of the events, we hope that more people than ever will have the chance to meet other users, share their experiences, talk to our developers, and hear about the latest technology.
I’ll be opening each event, and will take the opportunity to give some of the first public glimpses of Mathematica 11. There will be talks on machine learning, cloud computing, the Internet of Things, network analysis, and much more. The set of talks will be different at each event as developers and local users join us to present their insights, so check out the schedule for your local event.
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]]>Please allow me to introduce myself
I’m a man of precision and science
I’ve been around for a long, long time
Stole many a man’s pound and toise
And I was around when Louis XVI
Had his moment of doubt and pain
Made damn sure that metric rules
Through platinum standards made forever
Pleased to meet you
Hope you guess my name
In case you can’t guess: I am Jean-Charles de Borda, sailor, mathematician, scientist, and member of the Académie des Sciences, born on May 4, 1733, in Dax, France. Two weeks ago would have been my 283rd birthday. This is me:
In my hometown of Dax there is a statue of me. Please stop by when you visit. In case you do not know where Dax is, here is a map:
In Europe when I was a boy, France looked basically like it does today. We had a bit less territory on our eastern border. On the American continent, my country owned a good fraction of land:
I led a diverse earthly life. At 32 years old I carried out a lot of military and scientific work at sea. As a result, in my forties I commanded several ships in the Seven Years’ War. Most of the rest of my life I devoted to the sciences.
But today nobody even knows where my grave is, as my physical body died on February 19, 1799, in Paris, France, in the upheaval of the French Revolution. (Of course, I know where it is, but I can’t communicate it anymore.) My name is the twelfth listed on the northeast side of the Eiffel Tower:
Over the centuries many of my fellow Frenchman who joined me up here told me that I deserved a place in the Panthéon. But you will not find me there, nor at the Père Lachaise, Montparnasse, or Montmartre cemeteries.
But this is not why I still cannot rest in peace. I am a humble man; it is the kilogram that keeps me up at night. But soon I will be able to rest in peace at night for all time and approach new scientific challenges.
Let me tell you why I will soon find a good night’s sleep.
All my life, I was into mathematics, geometry, physics, and hydrology. And overall, I loved to measure things. You might have heard of substitution weighing (also called Borda’s method)—yes, this was my invention, as was the Borda count method. I also substantially improved the repeating circle. Here is where the story starts. The repeating circle was crucial in making a high-precision determination of the size of the Earth, which in turn defined the meter. (A good discussion of my circle can be found here.)
I lived in France when it was still a monarchy. Times were difficult for many people—especially peasants—partially because trade and commerce were difficult due to the lack of measures all over the country. If you enjoy reading about history, I highly recommend Kula’s Measures and Men to understand the weights and measurements situation in France in 1790. The state of the weights and measures were similar in other countries; see for instance Johann Georg Tralles’ report about the situation in Switzerland.
In August 1790, I was made the chairman of the Commission of Weights and Measures as a result of a 1789 initiative from Louis XVI. (I still find it quite miraculous that 1,000 years after Charlemagne’s initiative to unify weights and measures, the next big initiative in this direction would be started.) Our commission created the metric system that today is the International System of Units, often abbreviated as SI (le Système international d’unités in French).
In the commission were, among others, Pierre-Simon Laplace (think the Laplace equation), Adrien-Marie Legendre (Legendre polynomials), Joseph-Louis Lagrange (think Lagrangian), Antoine Lavoisier (conservation of mass), and the Marquis de Condorcet. (I always told Adrien-Marie that he should have some proper portrait made of him, but he always said he was too busy calculating. But for 10 years now, the politician Louis Legendre’s portrait has not been used in math books instead of Adrien-Marie’s. Over the last decades, Adrien-Marie befriended Jacques-Louis David, and Jacques-Louis has made a whole collection of paintings of Adrien-Marie; unfortunately, mortals will never see them.) Lagrange, Laplace, Monge, Condorcet, and I were on the original team. (And, in the very beginning, Jérôme Lalande was also involved; later, some others were as well, such as Louis Lefèvre‑Gineau.)
Three of us (Monge, Lagrange, and Condorcet) are today interred or commemorated at the Panthéon. It is my strong hope that Pierre-Simon is one day added; he really deserves it.
As I said before, things were difficult for French citizens in this era. Laplace wrote:
The prodigious number of measures in use, not only among different people, but in the same nation; their whimsical divisions, inconvenient for calculation, and the difficulty of knowing and comparing them; finally, the embarrassments and frauds which they produce in commerce, cannot be observed without acknowledging that the adoption of a system of measures, of which the uniform divisions are easily subjected to calculation, and which are derived in a manner the least arbitrary, from a fundamental measure, indicated by nature itself, would be one of the most important services which any government could confer on society. A nation which would originate such a system of measures, would combine the advantage of gathering the first fruits of it with that of seeing its example followed by other nations, of which it would thus become the benefactor; for the slow but irresistible empire of reason predominates at length over all national jealousies, and surmounts all the obstacles which oppose themselves to an advantage, which would be universally felt.
All five of the mathematicians (Monge, Lagrange, Laplace, Legendre, and Condorcet) have made historic contributions to mathematics. Their names are still used for many mathematical theorems, structures, and operations:
In 1979, Ruth Inez Champagne wrote a detailed thesis about the influence of my five fellow citizens on the creation of the metric system. For Legendre’s contribution especially, see C. Doris Hellman’s paper. Today it seems to me that most mathematicians no longer care much about units and measures and that physicists are the driving force behind advancements in units and measures. But I did like Theodore P. Hill’s arXiv paper about the method of conflations of probability distributions that allows one to consolidate knowledge from various experiments. (Yes, before you ask, we do have instant access to arXiv up here. Actually, I would say that the direct arXiv connection has been the greatest improvement here in the last millennium.)
Our task was to make standardized units of measure for time, length, volume, and mass. We needed measures that were easily extensible, and could be useful for both tiny things and astronomic scales. The principles of our approach were nicely summarized by John Quincy Adams, Secretary of State of the United States, in his 1821 book Report upon the Weights and Measures.
Originally we (we being the metric men, as we call ourselves up here) had suggested just a few prefixes: kilo-, deca-, hecto-, deci-, centi-, milli-, and the no-longer-used myria-. In some old books you can find the myria- units.
We had the idea of using prefixes quite early in the process of developing the new measurements. Here are our original proposals from 1794:
Side note: in my time, we also used the demis and the doubles, such as a demi-hectoliter (=50 liters) or a double dekaliter (=20 liters).
As inhabitants of the twenty-first century know, times, lengths, and masses are measured in physics, chemistry, and astronomy over ranges spanning more than 50 orders of magnitude. And the units we created in the tumultuous era of the French Revolution stood the test of time:
In the future, the SI might need some more prefixes. In a recent LIGO discovery, the length of the interferometer arms changed on the order of 10 yoctometers. Yoctogram resolution mass sensors exist. One yoctometer equals 10^{-24} meter. Mankind can already measure tiny forces on the order of zeptonewtons.
On the other hand, astronomy needs prefixes larger than 10^{24}. One day, these prefixes might become official.
I am a man of strict rules, and it drives me nuts when I see people in the twenty-first century not obeying the rules for using SI prefixes. Recently I saw somebody writing on a whiteboard that a year is pretty much exactly 𝜋 dekamegaseconds (𝜋 daMs):
While it’s a good approximation (only 0.4% off), when will this person learn that one shouldn’t concatenate prefixes?
The technological progress of mankind has occurred quickly in the last two centuries. And mega-, giga-, tera- or nano-, pico-, and femto- are common prefixes in the twenty-first century. Measured in meters per second, here is the probability distribution of speed values used by people. Some speeds (like speed limits, the speed of sound, or the speed of light) are much more common than others, but many local maxima can be found in the distribution function:
Here is the report we delivered in March of 1791 that started the metric system and gave the conceptual meaning of the meter and the kilogram, signed by myself, Lagrange, Laplace, Monge, and Concordet (now even available through what the modern world calls a “digital object identifier,” or DOI, like 10.3931/e-rara-28950):
Today most people think that base 10 and the meter, second, and kilogram units are intimately related. But only on October 27, 1790, did we decide to use base 10 for subdividing the units. We were seriously considering a base-12 subdivision, because the divisibility by 2, 3, 4, and 6 is a nice feature for trading objects. It is clear today, though, that we made the right choice. Lagrange’s insistence on base 10 was the right thing. At the time of the French Revolution, we made no compromises. On November 5, 1792, I even suggested changing clocks to a decimal system. (D’Alambert had suggested this in 1754; for the detailed history of decimal time, see this paper.) Mankind was not ready yet; maybe in the twenty-first century decimal clocks and clock readings would finally be recognized as much better than 24 hours, 60 minutes, and 60 seconds. I loved our decimal clocks—they were so beautiful. So it’s a real surprise to me today that mankind still divides the angle into 90 degrees. In my repeating circle, I was dividing the right angle into 100 grades.
We wanted to make the new (metric) units truly equal for all people, not base them, for instance, on the length of the forearm of a king. Rather, “For all time, for all people” (“À tous les temps, à tous les peuples”). Now, in just a few years, this dream will be achieved.
And I am sure there will come the day where Mendeleev’s prediction (“Let us facilitate the universal spreading of the metric system and thus assist the common welfare and the desired future rapprochement of the peoples. It will come not yet, slowly, but surely.”) will come true even in the three remaining countries of the world that have not yet gone metric:
The SI units have been legal for trade in the USA since the mid-twentieth century, when United States customary units became derived from the SI definitions of the base units. Citizens can choose which units they want for trade.
We also introduced the decimal subdivision of money, and our franc was in use from 1793 to 2002. At least today all countries divide their money on the basis of base 10—no coins with label 12 are in use anymore. Here is the coin label breakdown by country:
We took the “all” in “all people” quite seriously, and worked with our archenemy Britain and the new United States (through Thomas Jefferson personally) together to make a new system of units for all the major countries in my time. But, as is still so often the case today, politics won over reason.
I died on February 19, 1799, just a few months before the our group’s efforts. On June 22, 1799, my dear friend Laplace gave a speech about the finished efforts to build new units of length and mass before the new prototypes were delivered to the Archives of the Republic (where they are still today).
In case the reader is interested in my eventful life, Jean Mascart wrote a nice biography about me in 1919, and it is now available as a reprint from the Sorbonne.
Two of my friends, Jean Baptiste Joseph Delambre and Pierre Méchain, were sent out to measure distances in France and Spain from mountain to mountain to define the meter as one ten-millionth of the distance from the North Pole to the equator of the Earth. Historically, I am glad the mission was approved. Louis XVI was already under arrest when he approved the financing of the mission. My dear friend Lavoisier called their task “the most important mission that any man has ever been charged with.”
If you haven’t done so, you must read the book The Measure of All Things by Ken Adler. There is even a German movie about the adventures of my two old friends. Equipped with a special instrument that I had built for them, they did the work that resulted in the meter. Although we wanted the length of the meter to be one ten-millionth of the length of the half-meridian through Paris from pole to equator, I think today this is a beautiful definition conceptually. That the Earth isn’t quite as round as we had hoped for we did not know at the time, and this resulted in a small, regrettable error of 0.2 mm due to a miscalculation of the flattening of the Earth. Here is the length of the half-meridian through Paris, expressed through meters along an ellipsoid that approximates the Earth:
If they had elevation taken into account (which they did not do—Delambre and Méchain would have had to travel the whole meridian to catch every mountain and hill!), and had used 3D coordinates (meaning including the elevation of the terrain) every few kilometers, they would have ended up with a meter that was 0.4 mm too short:
Here is the elevation profile along the Paris meridian:
And the meter would be another 0.9 mm longer if measured with a yardstick the length of a few hundred meters:
Because of the fractality of the Earth’s surface, an even smaller yardstick would have given an even longer half-meridian.
It’s more realistic to follow the sea-level height. The difference between the length of the sea-level meridian meter and the ellipsoid approximation meter is just a few micrometers:
But at least the meridian had to go through Paris (not London, as some British scientists of my time proposed). But anyway, the meridian length was only a stepping stone to make a meter prototype. Once we had the meter prototype, we didn’t have to refer to the meridian anymore.
Here is a sketch of the triangulation carried out by Pierre and Jean Baptiste in their adventurous six-year expedition. Thanks to the internet and various French digitization projects, the French-speaking reader interested in metrology and history can now read the original results online and reproduce our calculations:
The part of the meridian through Paris (and especially through the Paris Observatory, marked in red) is today marked with the Arago markers—do not miss them during your next visit to Paris! François Arago remeasured the Paris meridian. After Méchain joined me up here in 1804, Laplace got the go-ahead (and the money) from Napoléon to remeasure the meridian and to verify and improve our work:
The second we derived from the length of a year. And the kilogram as a unit of mass we wanted to (and did) derive from a liter of water. If any liquid is special, it is surely water. Lavoisier and I had many discussions about the ideal temperature. The two temperatures that stand out are 0 °C and
4 °C. Originally we were thinking about 0 °C, as with ice water it is easy to see. But because of the minimal density of water at 4 °C, we later thought that would be the better choice. The switch to
4 °C was suggested by Louis Lefèvre-Gineau. The liter as a volume in turn we defined as one-tenth of a meter cubed. As it turns out, compared with high-precision measurements of distilled water,
1 kg equals the mass of 1.000028 dm^{3} of water. The interested reader can find many more details of the process of the water measurements here and about making the original metric system here. A shorter history in English can be found in the recent book by Williams and the ten-part series by Chisholm.
I don’t want to brag, but we also came up with the name “meter” (derived from the Greek metron and the Latin metrum), which we suggested on July 11 of 1792 as the name of the new unit of length. And then we had the area (=100 m^{2}) and the stere (=1 m^{3}).
And I have to mention this for historical accuracy: until I entered the heavenly spheres, I always thought our group was the first to carry out such an undertaking. How amazed and impressed I was when shortly after my arrival up here, I-Hsing and Nankung Yiieh introduced themselves to me and told me about their expedition from the years 721 to 725, more than 1,000 years before ours, to define a unit of length.
I am so glad we defined the meter this way. Originally the idea was to define a meter through a pendulum of proper length as a period of one second. But I didn’t want any potential change in the second to affect the length of the meter. While dependencies will be unavoidable in a complete unit system, they should be minimized.
Basing the meter on the Earth’s shape and the second on the Earth’s movement around the Sun seemed like a good idea at the time. Actually, it was the best idea that we could technologically realize at this time. We did not know how tides and time changed the shape of the Earth, or how continents drift apart. But we believed in the future of mankind, in ever-increasing measurement precision, but we did not know what concretely would change. But it was our initial steps for precisely measuring distances in France that were carried out. Today we have high-precision geo potential maps as high-order series of Legendre polynomials:
With great care, the finest craftsmen of my time melted platinum, and we forged a meter bar and a kilogram. It was an exciting time. Twice a week I would stop by Janety’s place when he was forging our first kilograms. Melting and forming platinum was still a very new process. And Janety, Louis XVI’s goldsmith, was a true master of forming platinum—to be precise, a spongelike eutectic made of platinum and arsenic. Just a few years earlier, on June 6, 1782, Lavoisier showed the melting of platinum in a hydrogen-oxygen flame to (the future) Tsar Paul I at a garden party at Versailles; Tsar Paul I was visiting Marie Antoinette and Loius XVI. And Étienne Lenoir made our platinum meter, and Jean Nicolas Fortin our platinum kilogram. For the reader interested in the history of platinum, I recommend McDonald’s and Hunt’s book.
Platinum is a very special metal; it has a high density and is chemically very inert. It is also not as soft as gold. The best kilogram realizations today are made from a platinum-iridium mixture (10% iridium), as adding iridium to platinum does improve its mechanical properties. Here is a comparison of some physical characteristics of platinum, gold, and iridium:
This sounds easy, but at the time the best scientists spent countless hours calculating and experimenting to find the best materials, the best shapes, and the best conditions to define the new units. But both the new meter bar and the new kilogram cylinder were macroscopic bodies. And the meter has two markings of finite width. All macroscopic artifacts are difficult to transport (we developed special travel cases); they change by very small amounts over a hundred years through usage, absorption, desorption, heating, and cooling. In the amazing technological progress of the nineteenth and twentieth centuries, measuring time, mass, and length with precisions better than one in a billion has become possible. And measuring time can even be done a billion times better.
I still vividly remember when, after we had made and delivered the new meter and the mass prototypes, Lavoisier said, “Never has anything grander and simpler and more coherent in all its parts come from the hands of man.” And I still feel so today.
Our goal was to make units that truly belonged to everyone. “For all time, for all people” was our motto. We put copies of the meter all over Paris to let everybody know how long it was. (If you have not done so, next time you visit Paris, make sure to visit the mètre étalon near to the Luxembourg Palace.) Here is a picture I recently found, showing an interested German tourist studying the history of one of the few remaining mètres étalons:
It was an exciting time (even if I was no longer around when the committee’s work was done). Our units served many European countries well into the nineteenth and large parts of the twentieth century. We made the meter, the second, and the kilogram. Four more base units (the ampere, the candela, the mole, and the kelvin) have been added since our work. And with these extensions, the metric system has served mankind very well for 200+ years.
How the metric system took off after 1875, the year of the Metre Convention, can be seen by plotting how often the words kilogram, kilometer, and kilohertz appear in books:
We defined only the meter, the seond, the liter, and the kilogram. Today many more name units belong to the SI: becquerel, coulomb, farad, gray, henry, hertz, joule, katal, lumen, lux, newton, ohm, pascal, siemen, sievert, tesla, volt, watt, and weber. Here is a list of the dimensional relations (no physical meaning implied) between the derived units:
Many new named units have been added since my death, often related to electrical and magnetic phenomena that were not yet known when I was alive. And although I am a serious person in general, I am often open to a joke or a pun—I just don’t like when fun is made of units. Like Don Knuth’s Potrzebie system of units, with units such as the potrzebie, ngogn, blintz, whatmeworry, cowznofski, vreeble, hoo, and hah. Not only are their names nonsensical, but so are their values:
Or look at Max Pettersson’s proposal for units for biology. The names of the units and the prefixes might sound funny, but for me units are too serious a subject to make fun of:
These unit names do not even rhyme with any of the proper names:
To reiterate, I am all in favor of having fun, even with units, but it must be clear that it is not meant seriously:
Or explicitly nonscientific units, such as helens for beauty, puppies for happiness, or darwins for fame are fine with me:
I am so proud that the SI units are not just dead paper symbols, but tools that govern the modern world in an ever-increasing way. Although I am not a comics guy, I love the recent promotion of the base units to superheroes by the National Institute of Standards and Technology:
Note that, to honor the contributions of the five great mathematicians to the metric system, the curves in the rightmost column of the unit-representing characters are given as mathematical formulas, e.g. for Dr. Kelvin we have the following purely trigonometric parametrization:
So we can plot Dr. Kelvin:
Having the characters in parametric form is handy: when my family has reunions, the little ones’ favorite activity is coloring SI superheroes. I just print the curves, and then the kids can go crazy with the crayons. (I got this idea a couple years ago from a coloring book by the NCSA.)
And whenever a new episode comes out, all us “measure men” (George Clooney, if you see this: hint, hint for an exciting movie set in the 1790s!) come together to watch it. As you can imagine, the last episode is our all-time favorite. Rumor has it up here that there will be a forthcoming book The Return of the Metrologists (2018 would be a perfect year) complementing the current book.
And I am glad to see that the importance of measuring and the underlying metric system is in modern times honored through the World Metrology Day on May 20, which is today.
In my lifetime, most of what people measured were goods: corn, potatoes, and other foods, wine, fabric, and firewood, etc. So all my country really needed were length, area, volume, angles, and, of course, time units. I always knew that the importance of measuring would increase over time. But I find it quite remarkable that only 200 years after I entered the heavenly spheres, hundreds and hundreds of different physical quantities are measured. Today even the International Organization for Standardization (ISO) lists, defines, and describes what physical quantities to use. Below is an image of an interactive Demonstration (download the notebook at the bottom of this post to interact with it) showing graphically the dimensions of physical quantities for subsets of selectable dimensions. First select two or three dimensions (base units). Then the resulting graphics show spheres with sizes proportional to the number of different physical quantities with these dimensions. Mouse over the spheres in the notebook to see the dimensions. For example, with “meter”, “second”, and “kilogram” checked, the diagram shows the units of physical quantities like momentum (kg^{1} m^{1} s^{-1}) or energy (kg^{2} m^{1} s^{-2}):
Here is a an excerpt of the code that I used to make these graphics. These are all physical quantities that have dimensions L^{2} M^{1} T^{-1}. The last one is the slightly exotic electrodynamic observable
:
Today with smart phones and wearable devices, a large number of physical quantities are measured all the time by ordinary people. “Measuring rules,” as I like to say. Or, as my (since 1907) dear friend William Thomson liked to say:
… when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.
Here is a graphical visualization of the physical quantities that are measured by the most common measurement devices:
Electrical and magnetic phenomena were just starting to become popular when I was around. Electromagnetic effects related to physical quantities that are expressed through the electric current only become popular much later:
I remember how excited I was when in the second half of the nineteenth century and the beginning of the twentieth century the various physical quantities of electromagnetism were discovered and their connections were understood. (And, not to be forgotten: the recent addition of memristance.) Here is a diagram showing the most important electric/magnetic physical quantities q_{k} that have a relation of the form q_{k}=q_{i} q_{j} with each other:
On the other hand, I was sure that temperature-related phenomena would soon be fully understood after my death. And indeed just 25 years later, Carnot proved that heat and mechanical work are equivalent. Now I also know about time dilation and length contraction due to Einstein’s theories. But mankind still does not know if a moving body is colder or warmer than a stationary body (or if they have the same temperature). I hear every week from Josiah Willard about the related topic of negative temperatures. And recently, he was so excited about a value for a maximal temperature for a given volume V expressed through fundamental constants:
For one cubic centimeter, the maximal temperature is about 5PK:
Long after my physical death, some of the giants of physics of the nineteenth century and early twentieth century, foremost among them James Clerk Maxwell, George Johnstone Stoney, and Max Planck (and Gilbert Lewis) were considering units for time, length, and mass that were built from unchanging properties of microscopic particles and the associated fundamental constants of physics (speed of light, gravitational constant, electron charge, Planck constant, etc.):
Maxwell wrote in 1870:
Yet, after all, the dimensions of our Earth and its time of rotation, though, relative to our present means of comparison, very permanent, are not so by any physical necessity. The earth might contract by cooling, or it might be enlarged by a layer of meteorites falling on it, or its rate of revolution might slowly slacken, and yet it would continue to be as much a planet as before.
But a molecule, say of hydrogen, if either its mass or its time of vibration were to be altered in the least, would no longer be a molecule of hydrogen.
If, then, we wish to obtain standards of length, time, and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wavelength, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules.
When we find that here, and in the starry heavens, there are innumerable multitudes of little bodies of exactly the same mass, so many, and no more, to the grain, and vibrating in exactly the same time, so many times, and no more, in a second, and when we reflect that no power in nature can now alter in the least either the mass or the period of any one of them, we seem to have advanced along the path of natural knowledge to one of those points at which we must accept the guidance of that faith by which we understand that “that which is seen was not made of things which do appear.’
At the time when Maxwell wrote this, I was already a man’s lifetime up here, and when I read it I applauded him (although at this time I still had some skepticism toward all ideas coming from Britain). I knew that this was the path forward to immortalize the units we forged in the French Revolution.
There are many physical constants. And they are not all known to the same precision. Here are some examples:
Converting the values of constants with uncertainties into arbitrary precision numbers is convenient for the following computations. The connection between the intervals and the number of digits is given as follows. The arbitrary precision number that corresponds to v ± δ is the number v with precision -log_{10}(2 δ/v) Conversely, given an arbitrary precision number (numbers are always convenient for computations), we can recover the v ± δ form:
After the exactly defined constants, the Rydberg constant with 11 known digits stands out for a very precisely known constant. On the end of the spectrum is G, the gravitational constant. At least once a month Henry Cavendish stops at my place with yet another idea on how to build a tabletop device to measure G. Sometimes his ideas are based on cold atoms, sometimes on superconductors, and sometimes on high-precision spheres. If he could still communicate with the living, he would write a comment to Nature every week. A little over a year ago Henry was worried that he should have done his measurements in winter as well in summer, but he was relieved to see that no seasonal dependence of G’s value seems to exist. The preliminary proposal deadline for the NSF’s Big G Challenge was just four days ago. I think sometime next week I will take a heavenly peek at the program officer’s preselected experiments.
There are more physical constants, and they are not all equal. Some are more fundamental than others, but for reasons of length I don’t want to get into a detailed discussion about this topic now. A good start for interested readers is Lévy-Leblond’s papers (also here), as well as this paper, this paper, and the now-classic Duff–Okun–Veneziano paper. For the purpose of making units from physical constants, the distinction of the various classes of physical constants is not so relevant.
The absolute values of the constants and their relations to heaven, hell, and Earth is an interesting subject on its own. It is a hot topic of discussion for mortals (also see this paper), as well as up here. Some numerical coincidences (?) are just too puzzling:
Of course, using modern mathematical algorithms, such as lattice reduction, we can indulge in the numerology of the numerical part of physical constants:
For instance, how can we form 𝜋 out of fundamental constant products?
Or let’s look at my favorite number, 10, the mathematical basis of the metric system:
And given a set of constants, there are many ways to form a unit of a given unit. There are so many physical constants in use today, you have to be really interested to keep up on them. Here are some of the lesser-known constants:
Physical constants appear in so many equations of modern physics. Here is a selection of 100 simple physics formulas that contain the fundamental constants:
Of course, more complicated formulas also contain the physical constants. For instance, the gravitational constant appears (of course!) in the formula of the gravitational potentials of various objects, e.g. for the potential of a line segment and of a triangle:
My friend Maurits Cornelis Escher loves these kinds of formulas. He recently showed me some variations of a few of his 3D pictures that show the equipotential surfaces of all objects in the pictures by triangulating all surfaces, then using the above formula—like his Escher solid. The graphic shows a cut version of two equipotential surfaces:
I frequently stop by at Maurits Cornelis’, and often he has company—usually, it is Albrecht Dürer. The two love to play with shapes, surfaces, and polyhedra. They deform them, Kelvin-invert them, everse them, and more. Albrecht also likes the technique of smoothing with gravitational potentials, but he often does this with just the edges. Here is what a Dürer solid’s equipotential surfaces look like:
And here is a visualization of formulas that contain c^{α}-h^{β}-G^{γ} in the exponent space α-β-γ. The size of the spheres is proportional to the number of formulas containing c^{α}·h^{β}·G^{γ}; mousing over the balls in the attached notebook shows the actual formulas. We treat positive and negative exponents identically:
One of my all-time favorite formulas is for the quantum-corrected gravitational force between two bodies, which contains my three favorite constants: the speed of light, the gravitational constants, and the Planck constant:
Another of my favorite formulas is the one for the entropy of a black hole. It contains the Boltzmann constant in addition to c, h, and G:
And, of course, the second-order correction to the speed of light in a vacuum in the presence of an electric or magnetic field due to photon-photon scattering (ignoring a polarization-dependent constant). Even in very large electric and magnetic fields, the changes in the speed of light are very small:
In my lifetime, we did not yet understand the physical world enough to have come up with the idea of natural units. That took until 1874, when Stoney proposed for the first time natural units in his lecture to the British Science Association. And then, in his 1906–07 lectures, Planck made use of the now-called Planck units extensively, already introduced in his famous 1900 article in Annalen der Physik. Unfortunately, both these unit systems use the gravitational constant G prominently. It is a constant that we today cannot measure very accurately. As a result, also the values of the Planck units in the SI have only about four digits:
These units were never intended for daily use because they are either far too small or far too large compared to the typical lengths, areas, volumes, and masses that humans deal with on a daily basis. But why not base the units of daily use on such unchanging microscopic properties?
(Side note: The funny thing is that in the last 20 years Max Planck again doubts if his constant h is truly fundamental. He had hoped in 1900 to derive its value from a semi-classical theory. Now he hopes to derive it from some holographic arguments. Or at least he thinks he can derive the value of h/k_{B} from first principles. I don’t know if he will succeed, but who knows? He is a smart guy and just might be able to.)
Many exact and approximate relations between fundamental constants are known today. Some more might be discovered in the future. One of my favorites is the following identity—within a small integer factor, is the value of the Planck constant potentially related to the size of the universe?
Another one is Beck’s formula, showing a remarkable coincidence (?):
But nevertheless, in my time we never thought it would be possible to express the height of a giraffe through the fundamental constants. But how amazed I was nearly ten years ago, when looking through the newly arrived arXiv preprints to find a closed form for the height of the tallest running, breathing organism derived by Don Page. Within a factor of two he got the height of a giraffe (Brachiosaurus and Sauroposeidon don’t count because they can’t run) derived in terms of fundamental constants—I find this just amazing:
I should not have been surprised, as in 1983 Press, Lightman, Peierls, and Gold expressed the maximal running speed of a human (see also Press’ earlier paper):
In the same spirit, I really liked Burrows’ and Ostriker’s work on expressing the sizes of a variety of astronomical objects through fundamental constants only. For instance, for a typical galaxy mass we obtain the following expression:
This value is within a small factor from the mass of the Milky Way:
But back to units, and fast forward another 100+ years to the second half of the twentieth century: the idea of basing units on microscopic properties of objects gained more and more ground.
Since 1967, the second has been defined through 9,192,631,770 periods of the light from the transition between the two hyperfine levels of the ground state of the cesium 133, and the meter has been defined since 1983 as the distance light travels in one second when we define the speed of light as the exact quantity 299,792,458 meters per second. To be precise, this definition is to be realized at rest, at a temperature of 0 K, and at sea level, as motion, temperature, and the gravitational potential influence the oscillation period and (proper) time. Ignoring the sea-level condition can lead to significant measurement errors; the center of the Earth is about 2.5 years younger than its surface due to differences in the gravitational potential.
Now, these definitions for the unit second and meter are truly equal for all people. Equal not just for people on Earth right now, but also for in the future and far, far away from Earth for any alien. (One day, the 9,192,631,770 periods of cesium might be replaced by a larger number of periods of another element, but that will not change its universal character.)
But if we wanted to ground all units in physical constants, which ones should we choose? There are often many, many ways to express a base unit through a set of constants. Using the constants from the table above, there are thirty (thirty!) ways to combine them to make a mass dimension:
Because of the varying precision of the constants, the combinations are also of varying precision (and of course, of different numerical values):
Now the question is which constants should be selected to define the units of the metric system? Many aspects, from precision to practicality to the overall coherence (meaning there is no need for various prefactors in equations to compensate for unit factors) must be kept in mind. We want our formulas to look like F = m a, rather than containing explicit numbers such as in the Thanksgiving turkey cooking time formulas (assuming a spherical turkey):
Or in the PLANK formula (Max hates this name) for the calculation of indicated horsepower:
Here in the clouds of heaven, we can’t use physical computers, so I am glad that I can use the more virtual Wolfram Open Cloud to do my calculations and mathematical experimentation. I have played for many hours with the interactive units-constants explorer below, and agree fully with the choices made by the International Bureau of Weights and Measures (BIPM), meaning the speed of light, the Planck constant, the elementary charge, the Avogadro constant, and the Boltzmann constant. I showed a preliminary version of this blog to Edgar, and he was very pleased to see this table based on his old paper:
I want to mention that the most popular physical constant, the fine-structure constant, is not really useful for building units. Just by its special status as a unitless physical quantity, it can’t be directly connected to a unit. But it is, of course, one of the most important physical constants in our universe (and is probably only surpassed by the simple integer constant describing how many spatial dimensions our universe has). Often various dimensionless combinations can be found from a given set of physical constants because of relations between the constants, such as c_{2}=1/(ε_{0} μ_{0}). Here are some examples:
But there is probably no other constant that Paul Adrien Maurice Dirac and I have discussed more over the last 32 years than the fine-structure constant α=e_{2}/(4 𝜋 ε_{0} ħ c). Although up here we meet with the Lord regularly in a friendly and productive atmosphere, he still refuses to tell us a closed form of α . And he will not even tell us if he selected the same value for all times and all places. For the related topic of the values of the constants chosen, he also refuses to discuss fine tuning and alternative values. He says that he chose a beautiful expression, and one day we will find out. He gave some bounds, but they were not much sharper than the ones we know from the Earth’s existence. So, like living mortals, for now we must just guess mathematical formulas:
Or guess combinations of constants:
And here is one of my favorite coincidences:
And a few more:
The rise in importance and usage of the physical constants is nicely reflected in the scientific literature. Here is a plot of how often (in publications per year) the most common constants appear in scientific publications from the publishing company Springer. The logarithmic vertical axis shows the exponential increase in how often physical constants are mentioned:
While the fundamental constants are everywhere in physics and chemistry, one does not see them so much in newspapers, movies, or advertisements, as they deserve. I was very please to see the introduction of the Measures for Measure column in Nature recently.
To give the physical constants the presence they deserve, I hope that before (or at least not long after) the redefinition we will see some interesting video games released that allow players to change the values of at least c, G, and h to see how the world around us would change if the constants had different values. It makes me want to play such a video game right now. With large values of h, not only could one build a world with macroscopic Schrödinger cats, but interpersonal correlations would also become much stronger. This could make the constants known to children at a young age. Such a video game would be a kind of twenty-first-century Mr. Tompkins adventure:
It will be interesting to see how quickly and efficiently the human brain will adapt to a possible life in a different universe. Initial research seems to be pretty encouraging. But maybe our world and our heaven are really especially fine-tuned.
The modern system of units, the current SI has, in addition to the second, the meter, and the kilogram, other units. The ampere is defined as the force between two infinitely long wires, the kelvin through the triple point of water, the mole through the kilogram and carbon-12, and the candela through blackbody radiation. If you have never read the SI brochure, I strongly encourage you to do so.
Two infinitely long wires are surely macroscopic and do not fulfill Maxwell’s demand (but it is at least an idealized system), and de facto it defines the magnetic constant. And the triple point of water needs a macroscopic amount of water. This is not perfect, but it’s OK. Carbon-12 atoms are already microscopic objects. Blackbody radiation is again an ensemble of microscopic objects, but a very reproducible one. So some of the current SI fulfills in some sense Maxwell’s goals.
But most of my insomnia over the last 50 years has been caused by the kilogram. It caused me real headaches, and sometimes even nightmares, when we could not put it on the same level as the second and the meter.
In the year of my physical death (1799), the first prototype of a kilogram, a little platinum cylinder, was made. About 39.7 mm in height and 39.4 mm in diameter, this was for 75 years “the” kilogram. It was made from the forged platinum sponge made by Janety. Miller gives a lot of the details of this kilogram. It is today in the Archives nationales. In 1879, Johnson Matthey (in Britain—the country I fought with my ships!), using new melting techniques, made the material for three new kilogram prototypes. Because of a slightly higher density, these kilograms were slightly smaller in size, at 39.14 mm in height. The cylinder was called KIII and became the current international prototype kilogram . Here is the last sentence from the preface of the mass determination of the the international prototype kilogram from 1885, introducing :
A few kilograms were selected and carefully compared to our original kilogram; for the detailed measurements, see this book. All three kilograms had a mass less than 1 mg different from the original kilogram. But one stood out: it had a mass difference of less than 0.01 mg compared to the original kilogram. For a detailed history of the making of , see Quinn. And so, still today, per definition, a kilogram is the mass of a small metal cylinder sitting in a safe at the International Bureau of Weights and Measures near Paris. (It’s technically actually not on French soil, but this is another issue.) In the safe, which needs three keys to be opened, under three glass domes, is a small platinum-iridium cylinder that defines what a kilogram is. For the reader’s geographical orientation, here is a map of Paris with the current kilogram prototype (in the southwest), our original one (in the northeast), both with a yellow border, and some other Paris visitor essentials:
In addition to being an artifact, it was so difficult to get access to the kilogram (which made me unhappy). Once a year, a small group of people checks if it is still there, and every few years its weight (mass) is measured. Of course, the result is, per definition and the agreement made at the first General Conference on Weights and Measures in 1889, exactly one kilogram.
Over the years the original kilogram prototype gained dozens of siblings in the form of other countries’ national prototypes, all of the same size, material, and weight (up to a few micrograms, which are carefully recorded). (I wish the internet had been invented earlier, so that I had a communication path to tell what happened with the stolen Argentine prototype 45; since then, it has been melted down.) At least, when they were made they had the same weight. Same material, same size, similarly stored—one would expect that all these cylinders would keep their weight. But this is not what history showed. Rather than all staying at the same weight, repeated measurements showed that virtually all other prototypes got heavier and heavier over the years. Or, more probable, the international prototype has gotten lighter.
From my place here in heaven I have watched many of these the comparisons with both great interest and concern. Comparing their weights (a.k.a. masses) is a big ordeal. First you must get the national prototypes to Paris. I have silently listened in on long discussions with TSA members (and other countries’ equivalents) when a metrologist comes with a kilogram of platinum, worth north of $50k in materials—and add another $20k for the making (in its cute, golden, shiny, special travel container that should only be opened in a clean room with gloves and mouth guard, and never ever touched by a human hand)—and explains all of this to the TSA. An official letter is of great help here. The instances that I have watched from up here were even funnier than the scene in the movie 1001 Grams.
Then comes a complicated cleaning procedure with hot water, alcohol, and UV light. The kilograms all lose weight in this process. And they are all carefully compared with each other. And the result is that with very high probability, “the” kilogram, our beloved international prototype kilogram (IPK), loses weight. This fact steals my sleep.
Here are the results from the third periodic verification (1988 to 1992). The graphic shows the weight difference compared to the international prototype:
For some newer measurements from the last two years, see this paper.
What I mean by “the” kilogram losing weight is the following. Per definition (independent of its “real objective” mass), the international prototype has a mass of exactly 1 kg. Compared with this mass, most other kilogram prototypes of the world seem to gain weight. As the other prototypes were made, using different techniques over more than 100 years, very likely the real issue is that the international prototype is losing weight. (And no, it is not because of Ceaușescu’s greed and theft of platinum that Romania’s prototype is so much lighter; in 1889 the Romanian prototype was already 953 μg lighter than the international prototype kilogram.)
Josiah Willard Gibbs, who has been my friend up here for more than 110 years, always mentions that his home country is still using the pound rather than the kilogram. His vote in this year’s election would clearly go to Bernie. But at least the pound is an exact fraction of the kilogram, so anything that will happen to the kilogram will affect the pound the same way:
But soon all my dreams and centuries-long hopes will come true and I can find sleep again. In 2018, two years from now, the greatest change in the history of units and measures since my work with my friend Laplace and the others will happen.
All units will be based on things that are accessible to everybody everywhere (assuming access to some modern physical instruments and devices).
The so-called new SI will reduce all of the seven base units to seven fundamental constants of physics or basic properties of microscopic objects. Down on Earth, they started calling them “reference constants.”
Some people also call the new SI quantum SI because of its dependence on the Planck constant h and the elementary charge e. In addition to the importance of the Planck constant h in quantum mechanics, the following two quantum effects are connecting h and e: the Josephson effect and its associated Josephson constant K_{J} = 2 e / h, and the quantum Hall effect with the von Klitzing constant R_{K} = h / e_{2}. The quantum metrological triangle: connecting frequency and electric current through a singe electron tunneling device, connecting frequency and voltage through the Josephson effect, and connecting voltage and electric current through the quantum Hall effect will be a beautiful realization of electric quantities. (One day in the future, as Penin has pointed out, we will have to worry about second-order QED effects, but this will be many years from now.)
The BIPM already has a new logo for the future International System of Units:
Concretely, the proposal is:
2. The meter will continue to be defined through an exactly defined speed of light.
3. The kilogram will be defined through an exactly defined value of the Planck constant.
4. The ampere will be defined through an exactly defined value of the elementary charge.
5. The kelvin will be defined through an exactly defined value of the Boltzmann constant.
6. The mole will be defined through an exact (counting) value.
7. The candela will be defined through an exact value of the candela steradian-to-watt ratio at a fixed frequency (already now the case).
I highly recommend a reading of the draft of the new SI brochure. Laplace and I have discussed it a lot here in heaven, and (modulo some small issues) we love it. Here is a quick word cloud summary of the new SI brochure:
Before I forget, and before continuing the kilogram discussion, some comments on the other units.
I still remember when we discussed introducing metric time in the 1790s: a 10-hour day, with 100 minutes per hour, and 100 seconds per minute, and we were so excited by this prospect. In hindsight, this wasn’t such a good idea. The habits of people are sometimes too hard to change. And I am so glad I could get Albert Einstein interested in the whole metrology over the past 50 years. We have had so many discussions about the meaning of time and that the second measures local time, and the difference between measurable local time and coordinate time. But this is a discussion for another day. The uncertainty of a second is today less than 10_{-16}. Maybe one day in the future, cesium will be replaced by aluminum or other elements to achieve 100 to 1,000 times smaller uncertainties. But this does not alter the spirit of the new SI; it’s just a small technical change. (For a detailed history of the second, see this article.)
Clearly, today’s definition of second is much better than one that depends on the Earth. At a time when stock market prices are compared at the microsecond level, the change of the length of a day due to earthquakes, polar melting, continental drift, and other phenomena over a century is quite large:
I have heard some chemists complain that their beloved unit, the mole, introduced into the SI only in 1971, will become trivialized. In the currently used SI, the mole relates to an actual chemical, carbon-12. In the new SI, it will be just a count of objects. A true chemical equivalent to a baker’s dozen, the chemist’s dozen. Based on the Avogadro constant, the mole is crucial in connecting the micro world with the macro world. A more down-to-Earth definition of the mole matters for such quantitative values—for example, pH values. The second is the SI base unit of time; the mole is the SI base unit of the physical quantity, or amount of substance:
But not everybody likes the term “amount of substance.” Even this year (2016), alternative names are being proposed, e.g. stoichiometric amount. Over the last decades, a variety of names have been proposed to replace “amount of substance.” Here are some examples:
But the SI system only defines the unit “mole.” The naming of the physical quantity that is measured in moles is up to the International Union of Pure and Applied Chemistry.
For recent discussions from this year, see the article by Leonard, “Why Is ‘Amount of Substance’ So Poorly Understood? The Mysterious Avogadro Constant Is the Culprit!”, and the article by Giunta, “What’s in a Name? Amount of Substance, Chemical Amount, and Stoichiometric Amount.”
Wouldn’t it be nice if we could have made a “perfect cube” (number) that would represent the Avogadro number? Such a representation would be easy to conceptualize. This was suggested a few years back, and at the time was compatible with the value of the Avogadro constant, and would have been a cube of edge length 84,446,888 items. I asked Srinivasa Ramanujan, while playing a heavenly round of cricket with him and Godfrey Harold Hardy, his longtime friend, what’s special about 84,446,888, but he hasn’t come up with anything deep yet. He said that 84,446,888=2^3*17*620933, and that 620,933 appears starting at position 1,031,622 in the decimal digits of 𝜋, but I can’t see any metrological relevance in this. With the latest value of the Avogadro constant, no third power of an integer number falls into the possible values, so no wonder there is nothing special.
Here is the latest CODATA (Committee on Data for Science and Technology) value from the NIST Reference on Constants, Units, and Uncertainty:
The candidate number 84,446,885 cubed is too small, and adding a one gives too large a number:
Interestingly, if we would settle for a body-centered lattice, with one additional atom per unit cell, then we could still maintain a cube interpretation:
A face-centered lattice would not work, either:
But a diamond (silicon) lattice would work:
To summarize:
Here is a little trivia:
Sometime amid the heights of the Cold War, the accepted value of the Avogadro constant suddenly changed in the third digit! This was quite a change, considering that there is currently a lingering controversy regarding the discrepancy in the sixth digit. Can you explain the sudden decrease in Avogadro constant during the Cold War?
Do you know the answer? If not, see here or here.
But I am diverting from my main thread of thoughts. As I am more interested in the mechanical units anyway, I will let my old friend Antoine Lavoisier judge the new mole definition, as he was the chemist on our team.
Josiah Willard Gibbs even convinced me that temperature should be defined mechanically. I am still trying to understand John von Neumann’s opinion on this subject, but because I never fully understand his evening lectures on type II and type III factors, I don’t have a firm opinion on the kelvin. Different temperatures correspond to inequivalent representations of the algebras. As I am currently still working my way through Ruetsche’s book, I haven’t made my mind up on how to best define the kelvin from an algebraic quantum field theory point of view. I had asked John on his opinion of a first-principle evaluation of h / k based on KMS states and Tomita–Takesaki theory, and even he wasn’t sure about it. He told me some things about thermal time and diamond temperature that I didn’t fully understand.
And then there is the possibility of deriving the value of the Boltzmann constant. Even 40 years after the Koppe–Huber paper, it is not clear if it is possible. It is a subject I am still pondering, and I am taking various options into account. As mentioned earlier, the meaning of temperature and how to define its units are not fully clear to me. There is no question that the new definition of the kelvin will be a big step forward, but I don’t know if it will be the end of the story.
This is one of the most direct, intuitive, and beautiful definitions in the new SI: the current is just the number of electrons that flow per second. Defining the value of the ampere through the number of elementary charges moved around is just a stroke of genius. When it was first suggested, Robert Andrews Millikan up here was so happy he invited many of us to an afternoon gathering in his yard. In practice (and in theoretical calculations), we have to exercise a bit more care, as we mainly measure the electric current of electrons in crystalline objects, and electrons are no longer “bare” electrons, but quasiparticles. But we’ve known since 1959, thanks to Walter Kohn, that we shouldn’t worry too much about this, and expect the charge of the electron in a crystal to be the same as the charge of a bare electron. As an elementary charge is a pretty small charge, the issue of measuring fractional charges as currents is not a practical one for now. I personally feel that Robert’s contribution to determining the value of the physical constants in the beginning of the twentieth century are not pointed out enough (Robert Andrews really knew what he was doing).
No, you will not get me started on my opinion the candela. Does it deserve to be a base unit? The whole story of human-centered physiological units is a complicated one. Obviously they are enormously useful. We all see and hear every day, even every second. But what if the human race continues to develop (in Darwin’s sense)? How will it fit together with our “for all time” mantra? I have my thoughts on this, but laying them out here and now would sidetrack me from my main discussion topic for today.
I also want to mention that originally I was very concerned about the introduction of some of the additional units that are in use today. In endless discussions with my chess partner Carl Friedrich Gauss here in heaven, he had originally convinced me that we can reduce all measurements of electric quantities to measurements of mechanical properties, and I already was pretty fluent in his CGS system, that originally I did not like it at all. But as a human-created unit system, it should be as useful as possible, and if seven units do the job best, it should be seven. In principle one could even eliminate a mass unit and express a mass through time and length. In addition to just being impractical, I strongly believe this is conceptually not the right approach. I recently discussed this with Carl Friedrich. He said he had the idea of just using time and length in the late 1820s, but abandoned such an approach. While alive, Carl Friedrich never had the opportunity to discuss the notion of mass as a synthetic a priori with Immanual, over the last century the two (Carl Friedrich and Immanuel) agreed on mass as an a priori (at least in this universe).
Our motto for the original metric system was, “For all time, for all people.” The current SI already realizes “for all people,” and by grounding the new SI in the fundamental constants of physics, the first promise “for all time” will finally become true. You cannot imagine what this means to me. If at all, fundamental constants seem to change maximally with rates on the order of 10_{-18} per year. This is many orders of magnitude away from the currently realized precisions for most units.
Granted, some things will get a bit numerically more cumbersome in the new SI. If we take the current CODATA values as exact values, then, for instance, the von Klitzing constant e^{2}/h will be a big fraction:
The integer part of the last result is, of course, 25,812Ω. Now, is this a periodic decimal fraction or a terminating fraction? The prime factorization of the denominator tells us that it is periodic:
Progress is good, but as happens so often, it comes at a price. While the new constant-based definitions of the SI units are beautiful, they are a bit harder to understand, and physics and chemistry teachers will have to come up with some innovative ways to explain the new definitions to pupils. (For recent first attempts, see this paper and this paper.)
And in how many textbooks have I seen that the value of the magnetic constant (permeability of the vacuum) μ_{0} is 4 𝜋 10^{-7} N / A^{2}? The magnetic and the electric constants will in the new SI become measured quantities with an error term. Concretely, from the current exact value:
With the Planck constant h exactly and the elementary charge e exactly, the value of μ_{0} would incur the uncertainty of the fine-structure constant α. Fortunately, the dimensionless fine-structure constant α is one of the best-known constants:
But so what? Textbook publishers will not mind having a reason to print new editions of all their books. They will like it—a reason to sell more new books.
With μ_{0} a measured quantity in the future, I predict one will see many more uses of the current underdog of the fundamental constant, the impedance of the vacuum Z in the future:
I applaud all physicists and metrologist for the hard work they’ve carried out in continuation of my committee’s work over the last 225 years, which culminated in the new, physical constant-based definitions of the units. So do my fellow original committee members. These definitions are beautiful and truly forever.
(I know it is a bit indiscreet to reveal this, but Joseph Louis Lagrange told me privately that he regrets a bit that we did not introduce base and derived units as such in the 1790s. Now with the Planck constant being too important for the new SI, he thought we should have had a named base unit for the action (the time integral over his Lagrangian). And then make mass a derived quantity. While this would be the high road of classical mechanics, he does understand that a base unit for the action would not have become popular with farmers and peasants as a daily unit needed for masses.)
I don’t have the time today to go into any detailed discussion of the quarterly garden fests that Percy Williams Bridgman holds. As my schedule allows, I try to participate in every single one of them. It is also so intellectually stimulating to listen to the general discussions about the pros and cons of alternative unit systems. As you can imagine, Julius Wallot, Jan de Boer, Edward Guggenheim, William Stroud, Giovanni Giorgi, Otto Hölder, Rudolf Fleischmann, Ulrich Stille, Hassler Whitney, and Chester Page are, not unexpectedly, most outspoken at these parties. The discussion about coherence and completeness of unit systems and what is a physical quantity go on and on. At the last event, the discussion of whether probability is or is not a physical quantity went on for six hours, with no decision at the end. I suggested inviting Richard von Mises and Hans Reichenbach the next time. They might have something to contribute. At the parties, Otto always complains that mathematicians do not care enough anymore about units and unit systems as they did in the past, and he is so happy to see at least theoretical physicists pick up the topic from time to time, like the recent vector-based differentiation of physical quantities or the recent paper on the general structure of unit systems. And when he saw in an article from last year’s Dagstuhl proceedings that modern type theory met units and physical dimensions, he was the most excited he had been in decades.
Interestingly, basically the same discussions came up three years ago (and since then regularly) in the monthly mountain walks that Claude Shannon organizes. Leo Szilard argues that the “bit” has to become a base unit of the SI in the future. In his opinion, information as a physical quantity has been grossly underrated.
Once again: the new SI will be just great! There are a few more details that I would like to see changed. The current status of the radian and the steradian, which SP 811 now defines as derived units, saying, “The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned.” But I see with satisfaction that the experts are discussing this topic recently quite in detail.
To celebrate the upcoming new SI here in heaven, we held a crowd-based fundraiser to celebrate this event. We raised enough funds to actually hire the master himself, Michelangelo. He will be making a sculpture. Some early sketches shown to the committee (I am fortunate to have the honorary chairmanship) are intriguing. I am sure it will be an eternal piece rivaling the David. One day every human will have the chance to see it (may it be a long time until then, dependent on your current age and your smoking habits). In addition to the constants and the units on their own, he plans to also work Planck himself, Boltzmann, and Avogadro into the sculpture, as these are the only three constants named after a person. Max was immediately accessible to model, but we are still having issues getting permission for Boltzmann to leave hell for a while to be a model. (Millikan and Fletcher were, understandably, a bit disappointed.) Ironically, it was Paul Adrien Maurice Dirac who came up with a great idea on how to convince Lucifer to get Boltzmann a Sabbath-ical. Ironically—because Paul himself is not so keen on the new SI because of the time dependence of the constants themselves over billions of years. But anyway, Paul’s clever idea was to point out that three fundamental constants, the Planck constant (6.62… × 10^{34} J · s), the Avogradro constant (6.02… × 10^{23} / mol), and the gravitational constant (6.6… × 10^{-11} m^{3} / (kg · s)) all start with the digit 6. And forming the number of the beast, 666, through three fundamental constants really made an impression on Lucifer, and I expect him to approve Ludwig’s temporary leave.
As an ex-mariner with an affinity for the oceans, I also pointed out to Lucifer that the mean ocean depth is exactly 66% of his height (2,443 m, according to a detailed re-analysis of Dante’s Divine Comedy). He liked this cute fact so much that he owes me a favor.
So far, Lucifer insists on having the combination G(m_{e} / (h k))^{1/2} on the sculpture. For obvious reasons:
We will see how this discussion turns out. As there is really nothing wrong with this combination, even if it is not physically meaningful, we might agree to his demands.
All of the new SI 2018 committee up here has also already agreed on the music, we will play Wojciech Kilar’s Sinfonia de motu, which uniquely represents the physical constants as a musical composition using only the notes c, g, e, h (b-flat in the English-speaking world), and a (where a represents the cesium atom). And we could convince Rainer Maria Rilke to write a poem for the event. Needless to say, Wojciech, who has now been with us for more than two years, agreed, and even offered to compose an exact version.
Down on Earth, the arrival of the constants-based units will surely also be celebrated in many ways and many places. I am looking forward especially to the documentary The State of the Unit, which will be about the history of the kilogram and its redefinition through the Planck constant.
As I already touched on, the most central point of the new SI will be the new definition of the kilogram. After all, the kilogram is the one artifact still present in the current SI that should be eliminated. In addition to the kilogram itself, many more derived units depend on it, say, the volt: 1 volt = 1 kilogram meters^{2}/(ampere second^{3}). Redefining the kilogram will make many (at least the theoretically inclined) electricians happy. Electrician have been using their exact conventional values for 25 years.
The value resulting from the convential value for the von Klitzing constant and the Josephson constant is very near to the latest CODATA value of the Planck constant:
A side note on the physical quantity that the kilogram represents: The kilogram is the SI base unit for the physical quantity mass. Mass is most relevant for mechanics. Through Newton’s second law, , mass is intimately related to force. Assume we have understood length and time (and so also acceleration). What is next in line, force or mass? William Francis Magie wrote in 1912:
It would be very improper to dogmatize, and I shall accordingly have to crave your pardon for a frequent expression of my own opinion, believing it less objectionable to be egotistic than to be dogmatic…. The first question which I shall consider is that raised by the advocates of the dynamical definition of force, as to the order in which the concepts of force and mass come in thought when one is constructing the science of mechanics, or in other words, whether force or mass is the primary concept…. He [Newton] further supplies the measurement of mass as a fundamental quantity which is needed to establish the dynamical measure of force…. I cannot find that Lagrange gives any definition of mass…. To get the measure of mass we must start with the intuitional knowledge of force, and use it in the experiments by which we first define and then measure mass…. Now owing to the permanency of masses of matter it is convenient to construct our system of units with a mass as one of the fundamental units.
And Henri Poincaré in his Science and Method says, “Knowing force, it is easy to define mass; this time the definition should be borrowed from dynamics; there is no way of doing otherwise, since the end to be attained is to give understanding of the distinction between mass and weight. Here again, the definition should be led up to by experiments.”
While I always had an intuitive feeling for the meaning of mass in mechanics, up until the middle of the twentieth century, I never was able to put it into a crystal-clear statement. Only over the last decades, with the help of Valentine Bargmann and Jean-Marie Souriau did I fully understand the role of mass in mechanics: mass is an element in the second cohomology group of the Lie algebra of the Galilei group.
Mass as a physical quantity manifests itself in different domains of physics. In classical mechanics it is related to dynamics, in general relativity to the curvature of space, and in quantum field theory mass occurs as one of the Casimir operators of the Poincaré group.
In our weekly “Philosophy of Physics” seminar, this year led by Immanuel himself, Hans Reichenbach, and Carl Friedrich von Weizsäcker (Pascual Jordan suggested this Dreimännerführung of the seminars), we discuss the nature of mass in five seminars. The topics for this year’s series are mass superselection rules in nonrelativistic and relativistic theories, the concept and uses of negative mass, mass-time uncertainty relations, non-Higgs mechanisms for mass generation, and mass scaling in biology and sports. I need at least three days of preparation for each seminar, as the recommended reading list is more than nine pages—and this year they emphasize the condensed matter appearance of these phenomena a lot! I am really looking forward to this year’s mass seminars; I am sure that I will learn a lot about the nature of mass. I hope Ehrenfest, Pauli, and Landau don’t constantly interrupt the speakers, as they did last year (the talk on mass in general relativity was particularly bad). In the last seminar of the series, I have to give my talk. In addition to metabolic scaling laws, my favorite example is the following:
I also intend to speak about the recently found predator-prey power laws.
For sports, I already have a good example inspired by Texier et al.: the relation between the mass of a sports ball and its maximal speed. The following diagram lets me conjecture speed_{max}~ln(mass). In the downloadable notebook, mouse over to see the sport, the mass of the ball, and the top speeds:
For the negative mass seminar, we had some interesting homework: visualize the trajectories of a classical point particle with complex mass in a double-well potential. As I had seen some of Bender’s papers on complex energy trajectories, the trajectories I got for complex masses did not surprise me:
End side note.
The complete new definition reads thus: The kilogram, kg, is the unit of mass; its magnitude is set by fixing the numerical value of the Planck constant to be equal to exactly 6.62606X*10^{-34} when it is expressed in the unit s^{-1} · m^{2} · kg, which is equal to J · s. Here X stands for some digits soon to be explicitly stated that will represent the latest experimental values.
And the kilogram cylinder can finally retire as the world’s most precious artifact. I expect soon after this event the international kilogram prototype will finally be displayed in the Louvre. As the Louvre had been declared “a place for bringing together monuments of all the sciences and arts” in May 1791 and opened in 1793, all of us on the committee agreed that one day, when the original kilogram was to be replaced with something else, it would end up in the Louvre. Ruling the kingdom of mass for more than a century, IPK deserves its eternal place as a true monument of the sciences. I will make a bet—in a few years the retired kilogram, under its three glass domes, will become one of the Louvre’s most popular objects. And the queue that physicists, chemists, mathematicians, engineers, and metrologists will form to see it will, in a few years, be longer than the queue for the Mona Lisa. I would also make a bet that the beautiful miniature kilogram replicas will within a few years become the best-selling item in the Louvre’s museum store:
At the same time, as a metrologist, maybe the international kilogram prototype should stay where it is for another 50 years, so that it can be measured against a post-2018 kilogram made from an exact value of the Planck constant. Then we would finally know for sure if the international kilogram prototype is/was really losing weight.
Let me quickly recapitulate the steps toward the new “electronic” kilogram.
Intuitively, one could have thought to define the kilogram through the Avogadro constant as a certain number of atoms of, say, ^{12}C. But because of binding energies and surface effects in a pile of carbon (e.g. diamond, graphene) made up from n = round(1 kg / m (^{12}C)) atoms to realize the mass of one kilogram, all the n carbon-12 atoms would have to be well separated. Otherwise we would have a mass defect (remember Albert’s famous E = m c^{2} formula), and the mass equivalent for one kilogram or compact carbon versus the same number of individual, well-separated atoms is on the order of 10^{-10}. Using the carbon-carbon bond energry, here is an estimation of the mass difference:
A mass difference of this size can for a 1 kg weight can be detected without problems with a modern mass comparator.
To give a sense of scale, this would be equivalent to the (Einsteinian) relativistic mass conversion of the energy expenditure of fencing for most of a day:
This does not mean one could not define a kilogram through the mass of an atom or a fraction of it. Given the mass of a carbon atom m (^{12}C), the atomic mass constant u = m (^{12}C) / 12 follows, and using u we can easily connect to the Planck constant:
I read with great interest the recent comparison of using different sets of constants for the kilogram definition. Of course, if the mass of a ^{12}C atom would be the defined value, then the Planck constant would become a measured, meaning nonexact, value. For me, having an exact value for the Planck constant is aesthetically preferable.
I have been so excited over the last decade following the steps toward the redefinition of the kilogram. For more than 20 years now, there has been a light visible at the end of the tunnel that would eliminate the one kilogram from its throne.
And when I read 11 years ago the article by Ian Mills, Peter Mohr, Terry Quinn, Barry Taylor, and Edwin Williams entitled “Redefinition of the Kilogram: A Decision Whose Time Has Come” in Metrologia (my second-favorite, late-morning Tuesday monthly read, after the daily New Arrivals, a joint publication of Hells’ Press, the Heaven Publishing Group, Jannah Media, and Deva University Press), I knew that soon my dreams would come true. The moment I read the Appendix A.1 Definitions that fix the value of the Planck constant h, I knew that was the way to go. While the idea had been floating around for much longer, it now became a real program to be implemented within a decade (give or take a few years).
James Clerk Maxwell wrote in his 1873 A Treatise on Electricity and Magnetism:
In framing a universal system of units we may either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science; or, if we expect soon to be able to determine the mass of a single molecule of a standard substance, we may wait for this determination before fixing a universal standard of mass.
Until around 2005, James Clerk thought that mass should be defined through the mass of an atom, but he came around over the last decade and now favors the definition through Planck’s constant.
In a discussion with Albert Einstein and Max Planck (I believe this was in the early seventies) in a Vienna-style coffee house (Max loves the Sachertorte and was so happy when Franz and Eduard Sacher opened their now-famous HHS (“Heavenly Hotel Sacher”)), Albert suggested using his two famous equations, E = m c^{2} and E = h f, to solve for m to get m = h f / c^{2}. So, if we define h as was done with c, then we know m because we can measure frequencies pretty well. (Compton was arguing that this is just his equation rewritten, and Niels Bohr was remarking that we cannot really trust E = m c^{2} because of its relatively weak experimental verification, but I think he was just mocking Einstein, retaliating for some of the Solvay Conference Gedankenexperiment discussions. And of course, Bohr could not resist bringing up Δm Δt ~ h / c^{2} as a reason why we cannot define the second and the kilogram independently, as one implies an error in the other for any finite mass measurement time. But Léon Rosenfeld convinced Bohr that this is really quite remote, as for a day measurement time this limits the mass measurement precision to about 10^{52} kg for a kilogram mass m.)
An explicit frequency equivalent f = m c^{2} / h is not practical for a mass of a kilogram as it would mean f ~ 1.35 10^{50} Hz, which is far, far too large for any experiment, dwarfing even the Planck frequency by about seven orders of magnitude. But some recent experiments from Berkeley from the last few years will maybe allow the use of such techniques at the microscopic scale. For more than 25 years now, in every meeting of the HPS (Heavenly Physical Society), Louis de Broglie insists on these frequencies being real physical processes, not just convenient mathematical tools.
So we need to know the value of the Planck constant h. Still today, the kilogram is defined as the mass of the IPK. As a result, we can measure the value of h using the current definition of the kilogram. Once we know the value of h to a few times 10^{-8} (this is basically where we are right now), we will then define a concrete value of h (very near or at the measured value). From then on, the kilogram will become implicitly defined through the value of the Planck constant. At the transition, the two definitions overlap in their uncertainties, and no discontinuities arise for any derived quantities. The international prototype has lost over the last 100 years on the order of 50 μg weight, which is a relative change of 5 × 10^{-8}, so a value for the Planck constant with an error less than 2 × 10^{-8} does guarantee that the mass of objects will not change in a noticeable manner.
Looking back over the last 116 years, the value of the Planck constant gained about seven digits in precision. A real success story! In his paper “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Max Planck for the first time used the symbol h, and gave for the first time a numerical value for the Planck constant (in a paper published a few month earlier, Max used the symbol b instead of h):
(I had asked Max why he choose the symbol h, and he said he can’t remember anymore. Anyway, he said it was a natural choice in conjunction with the symbol k for the Boltzmann constant. Sometimes one reads today that h was used to express the German word Hilfsgrösse (auxiliary helping quantity); Max said that this was possible, and that he really doesn’t remember.)
In 1919, Raymond Thayer Birge published the first detailed comparison of various measurements of the Planck constant:
From Planck’s value 6.55 × 10^{-34} J · s to the 2016 value 6.626070073(94) × 10^{-34} J · s, amazing measurement progress has been made.
The next interactive Demonstration allows you to zoom in and see the progress in measuring h over the last century. Mouse over the Bell curves (indicating the uncertainties of the values) in the notebook to see the experiment (for detailed discussions of many of the experiments for determining h, see this paper):
There have been two major experiments carried out over the last few years that my original group eagerly followed from the heavens: the watt balance experiment (actually, there is more than one of them—one at NIST, two in Paris, one in Bern…) and the Avogadro project. As a person who built mechanical measurements when I was alive, I personally love the watt balance experiment. Building a mechanical device that through a clever trick by Bryan Kibble eliminates an unknown geometric quantity gets my applause. The recent do-it-yourself LEGO home version is especially fun. With an investment of a few hundred dollars, everybody can measure the Planck constant at home! The world has come a long way since my lifetime. You could perhaps even check your memory stick before and after you put a file on it and see if its mass has changed.
But my dear friend Lavoisier, not unexpectedly, always loved the Avogadro project that determines the value of the Avogadro constant to high precision. Having 99.995% pure silicon makes the heart of a chemist beat faster. I deeply admire the efforts (and results) in making nearly perfect spheres out of them. The product of the Avogadro constant with the Planck constant N_{A} h is related to the Rydberg constant. Fortunately, as we saw above, the Rydberg constant is known to about 11 digits; this means that knowing N_{A} h to a high precision allows us to find the value of our beloved Planck constant h to high precision. In my lifetime, we started to understand the nature of the chemical elements. We knew nothing about isotopes yet—if you had told me that there are more than 20 silicon isotopes, I would not even have understood the statement:
I am deeply impressed how mankind today can even sort the individual atoms by their neutron count. The silicon spheres of the Avogadro project are 99.995 % silicon 28—much, much more than the natural fraction of this isotope:
While the highest-end beam balances and mass comparators achieve precisions of 10^{-11}, they can only compare masses but not realize one. Once the Planck constant has a fixed value using the watt balance, a mass can be constructively realized.
I personally think the Planck constant is one of the most fascinating constants. It reigns in the micro world and is barely visible at macroscopic scales directly, yet every macroscopic object holds together just because of it.
A few years ago I was getting quite concerned that our dream of eternal unit definitions would never be realized. I could not get a good night’s sleep when the value for the Planck constant from the watt balance experiments and the Avogadro silicon sphere experiments were far apart. How relieved I was to see that over the last few years the discrepancies were resolved! And now the working mass is again in sync with the international prototype.
Before ending, let me say a few words about the Planck constant itself. The Planck constant is the archetypal quantity that one expects to appear in quantum-mechanical phenomena. And when the Planck constant goes to zero, we recover classical mechanics (in a singular limit). This is what I myself thought until recently. But since I go to the weekly afternoon lectures of Vladimir Arnold, which he started giving in the summer of 2010 after getting settled up here, I now have strong reservations against such simplistic views. In his lecture about high-dimensional geometry, he covered the symplectic camel; since then, I view the Heisenberg uncertainty relations more as a classical relic than a quantum property. And since Werner Heisenberg recently showed me the Brodsky–Hoyer paper on ħ expansions, I have a much more reserved view on the BZO cube (the Bronshtein–Zelmanov–Okun cGh physics cube). And let’s not forget recent attempts to express quantum mechanics without reference to Planck’s constant at all. While we understand a lot about the Planck constant, its obvious occurrences and uses (such as a “conversion factor” between frequency and energy of photons in a vacuum), I think its deepest secrets have not yet been discovered. We will need a long ride on a symplectic camel into the deserts of hypothetical multiverses to unlock it. And Paul Dirac thinks that the role of the Planck constant in classical mechanics is still not well enough understood.
For the longest time, Max himself thought that in phase space (classical or through a Wigner transform), the minimal volume would be on the order of his constant h. As one of the fathers of quantum mechanics, Max follows the conceptual developments still today, especially the decoherence program. How amazed was he when sub-h structures were discovered 15 years ago. Eugene Wigner told me that he had conjectured such fine structures since the late 1930s. Since then, he has loved to play around with plotting Wigner functions for all kind of hypergeometric potentials and quantum carpets. His favorite is still the Duffing oscillator’s Wigner function. A high-precision solution of the time-dependent Schrödinger equations followed by a fractional Fourier transform-based Wigner function construction can be done in a straightforward and fast way. Here is how a Gaussian initial wavepacket looks after three periods of the external force. The blue rectangle is an area with in the x p plane of area h:
Here are some zoomed-in (colored according to the sign of the Wigner function) images of the last Wigner function. Each square has an area of 4 h and shows a variety of sub-Planckian structures:
For me, the forthcoming definition of the kilogram through the Planck constant is a great intellectual and technological achievement of mankind. It represents two centuries of hard work at metrological institutes, and cements some of the deepest physical truths found in the twentieth century into the foundations of our unit system. At once a whole slew of units, unit conversions, and fundamental constants will be known with greater precision. (Make sure you get a new CODATA sheet after the redefinition and have the pocket card with the new constant values with you always until you know all the numbers by heart!) This will open a path to new physics and new technologies. In case you make your own experiments determining the values of the constants, keep in mind that the deadline for the inclusion of your values is July 1, 2017.
The transition from the platinum-iridium kilogram, historically denoted , to the kilogram based on the Planck constant h can be nicely visualized graphically as a 3D object that contains both characters. Rotating it shows a smooth transition of the projection shape from to h representing over 200 years of progress in metrology and physics:
The interested reader can order a beautiful, shiny, 3D-printed version here. It will make a perfect gift for your significant other (or ask your significant other to get you one) for Christmas to be ready for the 2018 redefinition, and you can show public support for it as a pendent or as earrings. (Available in a variety of metals, platinum is, obviously, the most natural choice, and it is under $5k—but the $82.36 polished silver version looks pretty nice too.)
Here are some images of golden-looking versions of Toh3D (up here, gold, not platinum is the preferred metal color):
I realize that not everybody is (or can be) as excited as I am about these developments. But I see forward to the year 2018 when, after about 225 years, the kilogram as a material artifact will retire and a fundamental constant will replace it. The new SI will base our most important measurement standards on twenty-first century technology.
If the reader has questions or comments, don’t hesitate to email me at jeancharlesdeborda@gmail.com; based on recent advances in the technological implications of EPR=ER, we now have a much faster and more direct connection to Earth.
À tous les temps, à tous les peuples!
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In this blog we will show and briefly discuss some formerly unknown derivatives of special functions (primarily Bessel and related functions), and explore the history and current status of differentiation by parameters of hypergeometric and other functions. One of the main formulas found (more details below) is a closed form for the first derivative of one of the most popular special functions, the Bessel function J:
Many functions of mathematical physics (i.e. functions that are used often and therefore have special names) depend on several variables. One of them is usually singled out and designated as the “argument,” while others are usually called “parameters” or sometimes “indices.” These special functions can have any number of parameters. For example (see the Wolfram Functions Site), the Bessel functions J_{v}(z) and I_{v}(z), the Neumann function Y_{v}(z), the Macdonald function K_{v}(z), and the Struve functions H_{v}(z) and L_{v}(z) take only one parameter (called the index), while the Whittaker functions M_{μ,v}(z) and W_{μ,v}(z) as well as the confluent hypergeometric functions _{1}F_{1}(a; b; z) and U (a, b, z) take two parameters. The Anger functions J_{v}(z) and as well as the Weber functions E_{v}(z) and can have one or two parameters (in the cases of two parameters, they are called generalized Anger and Weber functions). The Appell and Humbert functions mostly have from three to five parameters, while more complicated special functions such as the generalized hypergeometric function _{p}F_{q}(a_{1}, …, a_{p}; b_{1}, …, b_{q}; z) can have any finite number of parameters.
Among other properties, differentiation of special functions plays an essential role, since derivatives characterize the behavior of functions when these variables change, and they are also important for studying the differential equations of these functions. Usually, differentiation of a special function with respect to its argument presents no essential difficulties. The largest collection of such derivatives, comprising the first, second, symbolic, and even fractional order for 200+ functions, can be found in the section “Differentiation” at the Wolfram Functions Site (for example, see this section, which includes 21 such derivatives for the Bessel function J_{v}(z)), or Y. A. Brychkov’s Handbook of Special Functions). The majority of these formulas are also available directly in the Wolfram Language through the use of the built-in symbols MathematicalFunctionData and EntityValue.
Derivatives with respect to parameters (as distinct from the argument), however, can generally be much more difficult to compute; that is the subject of this blog. Remarkably, the formula above, involving the generic first-order (with respect to the single parameter v) derivative of one of the most commonly occurring special functions in mathematical physics, has only just been discovered in closed form, and this perhaps surprising fact speaks to the difficulty of the general problem. So, using the Bessel J function as a characteristic example, let us take a brief walking tour through the history of special function differentiation.
Often, people even well acquainted with calculus tend to think that integration is difficult and differentiation is easy. The folklore is that “differentiation is mechanics, integration is art.” But the spirit of this saying is true only for elementary functions, where differentiation again produces elementary functions (or combinations thereof). For hypergeometric and other lesser-known special functions, when differentiation is carried out with respect to parameters, it can typically produce complicated functions of a more general class.
The distinction between differentiation with respect to parameters versus differentiation with respect to argument is exemplified by the Bessel J function. The derivative of Bessel J with respect to its argument z has been known for quite some time, and has this relatively simple closed form:
However, the analytic evaluation of its derivative with respect to parameters (e.g. v in the above equation) is more complicated. Often, derivatives such as these can be written in the form of an integral or infinite series, but those objects cannot be represented in closed form through other simple or well-known functions. Historically, some special functions were introduced for the sole purpose of giving a simple notation for the derivatives of other, more basic functions. For example, the polygamma function arose in this way as a means of representing derivatives of the gamma function.
The generalized hypergeometric function _{p}F_{q}(a_{1}, …, a_{p}; b_{1}, …, b_{q}; z) and its derivatives play an essential role in the solution of various problems in theoretical and applied mathematics (see, for example, this article by L. U. Ancarani and G. Gasaneo concerning the applications of derivatives by parameters in quantum mechanics). The generalized hypergeometric function generates as special cases many of the most-used elementary functions (e.g. the trigonometric, hyperbolic, logarithmic, and inverse trigonometric functions) as well as many families of more specialized functions, including the Bessel, Struve, Kelvin, Anger–Weber, incomplete gamma, and integral (exponential, sine, and cosine) functions. In the case p = 0, q = 1, the generalized hypergeometric function _{p}F_{q} subsumes the Bessel family of functions J_{v}(z), I_{v}(z), Y_{v}(z), and K_{v}(z). To be precise, Bessel J, for example, has the following hypergeometric representation:
Interestingly, the history of the function J_{v}(z) starts nearly exactly 200 years ago. In the 1816–17 proceedings of the Berlin Academy (published in 1819), in the work Analytische Auflösung der Keplerschen Aufgabe, Friedrich Wilhelm Bessel deals with the so-called Kepler equation
M=E-e sin(E), where M is the mean anomaly, E is the eccentric anomaly, and e is the eccentricity of a Keplerian orbit. The solution of this equation can be represented (in today’s notation) through Bessel functions of integer order:
In this first work, Bessel does not yet use the modern notation, but his function appears already implicitly. For example, he uses the following sum (note that Bessel uses Gauss’ notation 𝜋_{i} for i!):
In modern times, we could write this as the sum of two Bessel functions, which can be shown in the Wolfram Language:
Furthermore, this sum is just the first derivative of the single Bessel function -2 a e J(i, e i):
In a second work from 1824, Bessel uses the nearly modern notation (with J↔I) to denote his function:
He also derives fundamental relations for this function:
Various special instances of the generic Bessel function occur already in the writings of Bernoulli, Euler, d’Alembert, and others (see this article for a detailed account). The main reference about Bessel functions today is still the classical monograph by G. N. Watson, A Treatise on the Theory of Bessel Functions, which has been republished and extended many times since 1922.
So while the derivatives of Bessel J with respect to the argument z were known since the beginning of the nineteenth century, it took until the middle of the twentieth century before special cases for derivatives with respect to the index v were found. The derivatives of some Bessel functions with respect to the parameter v at the points v ==0, 1, 2,… and v == 1/2 were obtained by J. R. Airey in 1935, and the expressions for other Bessel family functions were given by W. Magnus, F. Oberhettinger, and R. P. Soni in “Formulas and Theorems for the Special Functions of Mathematical Physics” (1966):
Generalizations to any half-integer values of v were reported more recently at an international conference on abstract and applied analysis (Hanoi, 2002) as the following:
These results, along with expressions for the parametric derivatives of Struve functions at integer and half-integer values, were published in 2004–2005. Various new formulas for differentiation with respect to parameters of the Anger and Weber functions, Kelvin functions, incomplete gamma functions, parabolic cylinder functions, Legendre functions, and the Gauss, generalized, and confluent hypergeometric functions can be found in the Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. For a short survey and references, see H. Cohl.
But perhaps amazingly, given all this work, the first derivatives of the Bessel functions in closed form for arbitrary values of the parameter were obtained only in 2015 (Y. A. Brychkov, “Higher Derivatives of the Bessel Functions with Respect to the Order” (2016)). They are expressed as combinations of products of Bessel functions and generalized hypergeometric functions; for example:
The plots below give some impressions about the behavior of the Bessel function J_{v}(z) and its derivative on the domains of interest. First, we plot (in the real v-z plane) the expression giving the first derivative of J_{v}(z) with respect to v (see the first equation of this article):
For a fixed index, specifically v = 𝜋, we show the Bessel function J, together with its first two derivatives:
It is interesting to note that the first two derivatives (with respect to z and with respect to v) have nearly the same zeros.
It is remarkable that even almost 300 years after the introduction of a classical function (the Bessel function J_{0}(z) was introduced by Daniel Bernoulli in 1732), it is still possible to find new and relatively simple formulas relating to such functions. The actual derivation of the formula introduced above for (along with the related results for , and the Neumann, Macdonald, and Kelvin functions) was complicated, and was achieved using the help of the Wolfram Language. Details of the derivation are published, and here we give a rough sketch of derivation of the approach that was used.
First, we recall that the Bessel and other functions in which we are interested for this program are of the hypergeometric type; differentiation by parameters of the generic hypergeometric function of a single variable _{p}F_{q}(a_{1}, …, a_{p}; b_{1}, …, b_{q}; z) requires more complicated functions of the hypergeometric type with more than one variable (see this article by L. U. Ancarani and G. Gasaneo). The first derivative with respect to an upper “parameter” a_{k}, and all derivatives of symbolic integer order m with respect to a “lower” parameter b_{k} of the generalized hypergeometric function, can be expressed in terms of the Kampé de Fériet hypergeometric function of two variables by the following formulas:
Above, the Kampé de Fériet hypergeometric function is defined by the double series (see defining expressions here and here).
The Kampé de Fériet function can be considered as a generalization of the hypergeometric function to two variables:
A corresponding regularized version of the function can also be defined by replacing the product of Pochhammer symbols in the denominator with
The Kampé de Fériet function can be used to obtain a representation of the derivatives of the Bessel function J with respect to its parameter:
This expression coincides with the simpler formula above, which involves hypergeometric functions of one variable, though this is not necessarily easy to see (we don’t yet have a fully algorithmic treatment for the reduction of multivariate hypergeometric functions into expressions containing only univariate hypergeometric functions, and this has contributed to the difficulty in discovering formulas like the one discussed here).
Double series, like the one given above defining the generalized hypergeometric functions of two variables, also arise in the evaluation of Mellin transforms of products of three Meijer G-functions:
The right side of this formula includes a Meijer G-function of two variables that generically can be represented, in the non-logarithmic case, as a finite sum of Kampé de Fériet hypergeometric functions with some coefficients, by analogy with these two formulas. Finally, the Kampé de Fériet function also arises in the separation of the real and imaginary parts of hypergeometric functions of one variable, z==x+ⅈy, with real arguments:
It should be mentioned that in recent years the hypergeometric functions of many variables have found growing applications in the realms of quantum field theory, chemistry, engineering, and, in particular, communication theory and radio location. Many quite practical results can be represented using such functions, and consequently, most of the principal results in this field are obtained in the applied science literature. The theory of such functions in theoretical mathematics circles has thus far been elaborated relatively weakly.
We are lucky here at Wolfram to have the originator of these new and exciting symbolic derivative formulas, Yury Brychkov, as part of our team, enabling us to bring this constantly developing area of research work into the grasp of our users. We are also lucky to have at our disposal the Entity framework of the Wolfram Language, which allows, among other things, for the integration of cutting-edge new results such as these on the timescale of weeks or days, in a computable format, easily accessible from a variety of Wolfram Language platforms. For example, in Mathematica, one can evaluate the following:
This obtains the principal formula of this article. We can attempt to confirm the formula numerically by first substituting global values of v and z and activating; we get:
Next, we separate the left- and right- hand sides (to allow for floating-point numerical errors) and substitute random values for the argument and parameter to obtain:
The numerical derivative of the left-hand side is computed internally in the Wolfram Language via a limiting procedure. The equality of left- and right-hand sides, and therefore the correctness of the original derivative formula, is thus apparent.
Aside from the many new results for symbolic and parametric derivatives alluded to in this article and available only through EntityValue (though deeper integration into future versions of the Wolfram Language is an ongoing effort), a large number of long-standing results in this field have already been implemented into the Mathematica kernel and the core Wolfram Language. By reason of their complexity, such derivatives by parameters are not evaluated automatically, but can be seen using the FunctionExpand command. For example:
At the second and higher order, derivatives of Bessel and related functions can still be expressed in terms of the Kampé de Fériet hypergeometric function of two variables, but the resulting formulas can be rather complicated, and can include the Bell Y polynomials:
The latter arise from expressing the Bessel function J_{v}(z) as the composition of the function
_{0}F_{1}(; v+1; w) and function
We utilize Faà di Bruno’s formula, which describes the nth derivative of a composition of m functions f_{i}(z)/; 1 ≤ i ≤ m. In the m=2 case (see here and here), we obtain, for instance, an expression involving the following:
The corresponding formula for generic m and n can be obtained and verified in the Wolfram Language as:
While the Bell Y’s—for which no generic closed forms exist—are generally needed to express higher-order derivatives, as this blog was headed to press one of the authors, Yury Brychkov, even found a way to eliminate Bell Y from the nth derivatives with respect to the parameter of the Bessel functions, leaving us with the remarkable
For the convenience of interested users who would like to see in one place all known formulas for derivatives of special functions with respect to parameters (including those listed above), we have collected and presented these formulas in the following ways:
In our next blog, we will continue presenting closed forms for derivatives, for a collection for 400+ functions with generic rules for derivatives of symbolic and fractional order. In the meantime, we hope you enjoy exploring on your own the world of special function differentiation in the Wolfram Language!
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]]>Programming Lab was conceived and designed with teaching in mind, and to help make Programming Lab the best possible learning environment, we’ve developed some new tools for both students and teachers. We invite you to preview these new materials at a special virtual event, New Resources for the Classroom: Virtual Workshop for Educators.
Programming Lab is built on two major components utilizing two different learning styles to teach coding in the Wolfram Language—Explorations are based on a jump-right-in-and-explore approach, and Stephen Wolfram’s book An Elementary Introduction to the Wolfram Language provides the basis for a systematic approach. We are introducing new educator-focused functionality for both of these components.
First are the Explorations. These are bite-sized coding exercises designed to create something or answer a specific question. We are enhancing these by introducing Teacher’s Editions, which provide goals, procedures, and helpful comments for each step within an Exploration. In addition to these, we have also developed “Programming Lab modules.” These are handy planning guides that are tailor-made to fit a specific duration: a three-day introduction, a five-day series, and so on. They range across topics that include astronomy, geography, math, and many others. These modules provide educators with the material they need to present Explorations that will complement the curriculum, or as a fun way to introduce a topic while simultaneously adding a programming language to their toolkits.
Explorations work well as targeted activities for students, but some students will want a more rigorous framework for learning the Wolfram Language. Those students will benefit from working through Stephen Wolfram’s An Elementary Introduction to the Wolfram Language, which is built into Programming Lab. At the end of each section of the book, there are exercises to help solidify your understanding. We’re enhancing these exercises by providing instant, automated grading of your answers.
To preview these new materials and to be the first to try them out, please join us at the virtual event on May 17, 2016, from 4–5pm US EDT (8–9pm GMT). All are welcome, as no programming experience is necessary! This event is part of a series of workshops for educators, ranging in focus from how to use Programming Lab to teaching computational thinking principles. Recordings of past events are available online. You can register for the latest educator workshop here.
]]>Stippling is a kind of drawing style using only points to mimic lines, edges, and grayscale. The entire drawing consists only of dots on a white background. The density of the points gives the impression of grayscale shading.
Back in 1510, stippling was first invented as an engraving technique, and then became popular in many fields because it requires just one color of ink.
Here is a photo of a fine example taken from an exhibition of lithography and copperplate art (the Centenary of European Engraving Exhibition held at the Hubei Museum of Art in March 2015; in case you’re curious, here is the museum’s official page in English).
The art piece is a lithographic print. From a view of one meter away, it shows remarkable detail and appears realistic. However, looking at it from a much closer distance, you can see it’s made up of hundreds of thousands of handcrafted stipples, with the original marble stone giving it even more texture.
From my point of view, this artistic stippling method is like a simulation of variable solidity using small dotted patterns, like a really dense dot matrix printer, except these patterns are handmade. It is fascinating looking at lithographs, observing how a macro reality emerges from uncountable random dots: the mirror reflection on the floor, the soft sense of the cloths, and the mottling on the building engraved by history. (Photos were taken from the same exhibition.)
As a technique invented five hundred years ago, stippling has become popular in so many fields way beyond its original intention because of its simplicity. It was once the standard for technical illustrations, especially in math and mechanics books. Even today, many people are interested in it. Experts from different domains are still using stippling in daily work.
Typical stippling almost always uses one or two colors. Actually, one can create all sorts of textures and tones without introducing artifacts—one cannot achieve both with techniques like hatching or line drawing—just by varying the density and distribution of the dots. That makes stippling an excellent choice when archaeologists, geologists, and biologists are doing fieldwork and want to record things they find immediately. They don’t want to travel along with heavy painting gear; a single pen is good enough for stippling. Even in modern days, compared to photography, stippling still holds some advantages despite requiring a large amount of time. For example, the author can mark and emphasize any features he or she wants right on the drawing; thus, the technique is an important skill—one that is still being taught and researched today.
Here to support my point, I made these dinnertime paper-napkin stipplings (if they look like a complete disaster to you, please forgive me, as I’m not an expert and they only took about five minutes):
It is also very interesting to note that stippling is such a natural way to distribute dense points that the strategy is actually adopted by many living things, like this one in my dish:
In the computer age, there is a special computer graphics domain called non-photorealistic rendering. Stippling is one of its basic artistic painting styles, and can be used as the foundation of many other synthetic styles, such as mosaic, stained glass, hatching, etc.
Back to the art side. Stippling shows interesting connections with Divisionism and Pointillism, which both belong to Neo-Impressionism.
In popular art, people are still attracted to this ancient art. There are even stipple tattoos on Pinterest. You can watch people doing it with stunning effect. Perhaps you can even learn stippling yourself on the internet.
Needless to say, stippling as a handcrafted art really requires lots of patience. The manual process is time consuming. Moreover, it takes a lot of practice to become skillful.
After trying so hard on the napkin without satisfaction, I had a closer look at the masterpieces. I noticed that the points in the drawings seem random but are actually ordered, a bit like the structure seen in a quasicrystal:
In the above comparison, the left figure corresponds to randomly distributed points, while the right figure corresponds to a distribution like those found in stippling drawings. Obviously, they are very different from each other.
It turns out when using points to approximate grayscale shading, a random set of points with uniform distribution is usually not good enough.
To illustrate that, both images in the comparison below have 63,024 points sampled from the same distribution with the same contrast. If we think of the original grayscale image as a two-dimensional distribution, the local density of the two resulting images are the same for any corresponding position. Therefore, the contrast of the two images must be the same as well, which can be illustrated by resizing them to a small enough scale so the detail of the points won’t distract our eyes.
Nevertheless, the one on the right is a good stippling (or, as it’s usually called, well spaced), while the one on the left has too many unwanted small artifacts—clumps and voids that do not exist in the original image.
Now it is not very hard to see that in the stippling graphics, the triangle formed with any three points nearest to each other is nearly equilateral. This turns out to be the essential property of a “good” stippling.
Given that unique micro scale character, I couldn’t help wondering: is it possible to generate a stippling drawing from any image programmatically?
The answer is yes—as I have just shown one of the results from my generator, and which I will describe in detail in the rest of this blog.
Actually, there are not only lots of papers on this topic, but also well-made applications.
Back to the first comparison example. In order to quantitatively reveal the “well-spaced” property, I draw the DelaunayMesh of the points and observe them:
With the meshes, I can confirm my equilateral triangle conjecture numerically by graphing the distribution of the interior angles of the cells in the meshes:
Recalling the duality between a Delaunay mesh and a Voronoi mesh, and with the help of lots of academic papers, I eventually inferred that my equilateral Delaunay mesh corresponds to the so-called centroidal Voronoi diagram (CVD). Indeed, the CVD is the de facto method for computer-generated stippling. Moreover, there is a dedicated algorithm for it due to Lloyd:
Here the key steps are the Voronoi diagram generation and the centroid finding. The former is a perfect match for the bounded version of the built-in function VoronoiMesh. The latter, as the Voronoi cells for a closed region are always closed convex polygons, has a simple formula that I’d like to briefly describe as follows for completeness. If you’re not into math, it can be skipped safely without harming the joy of stippling!
Now suppose I have a cell defined by n vertices ordered clockwise (or counterclockwise) {P_{1}=(x_{1}, y_{1}), P_{2}=(x_{2}, y_{2}),… P_{n}=(x_{n}, y_{n})}; then its centroid C can be determined as:
As a test, we generate 2,500 uniformly distributed random points in a square region [-1,1]×[-1,1]:
Its Voronoi diagram is:
Lloyd’s algorithm can be expressed as the following findCentroid function:
Here, just to illustrate the principle of the algorithm, and to make things simpler to understand by ensuring that the code structure is similar to the weighted centroidal Voronoi diagram (which I will describe later), I have defined my own findCentroid function. Note that for uniform cases, there is the much more efficient built-in function, MeshCellCentroid:
Each Polygon from the Voronoi mesh can be extracted with MeshPrimitives[...,2], which then should be piped to the findCentroid to complete one iteration:
Now we’re ready to animate the first 50 iteration results to give a rough illustration of the CVD:
There are various ways to show the difference between the point distributions before and after the process:
For example, I can use NearestNeighborGraph to illustrate their connectivity, which will highlight the unwanted voids in the former case:
Alternatively, as was shown earlier, I can compare the statistics of the interior angles. After Lloyd’s algorithm, the mesh angles are much nearer to 60°:
Finally, to give another intuitive impression on the “well-spaced” property from a different point of view, I’d like to compare the discrete Fourier transform of the initial points with the one of the refined points:
Basically, the Fourier transform counts the significances of periodic structures with different periods. The ones with the largest periods (say, a structure repeating every 100 pixels) correspond to the center-most positions, and the ones with the smallest periods (say, a structure repeating every 2 pixels) correspond to the furthest positions. The value at a position indicates the significance of the corresponding structure. In the above plots, the larger values are illustrated with whiter colors. The center white spots correspond to the sum of all the data, which is not of interest here. In the initial case, the significances of different periodic structures are almost the same; this indicates that it is a uniform random distribution (or so-called white noise). The refined case is more interesting. The large-scale periodic structures are all flattened out, illustrated as a black disk region around the center spot, which means the points distribute uniformly on the macro scale. Then there is a bright circle around the dark disk region. The circle shape indicates that the positions in that circle correspond to structures with nearly the same period but in different directions, which are the approximately equilateral triangles in the mesh with nearly the same size but random directions. The outer “rings” correspond to high-order structures, which should not appear in a perfect refined case. So it can be clearly seen that the refined points’ case effectively achieves both isotropism and low-stop filter. This is another facet of view to understand the term “well spaced.” Now I can say that the property is also related to the concept of the “blue color” of noise.
So far, it seems I have solved the puzzle of stippling a uniform area, but how about a non-uniform area—say, the Lena example?
After some trials, I realized Lloyd’s algorithm could not be naively adapted here, as it always smoothed out the distribution and resulted in a uniform CVD. It looked like I had to find another way.
The answer turned out to be a simple modification of Lloyd’s algorithm. However, back when I thought about the problem, a totally different idea showed an interesting possibility. It’s a mathematically beautiful method, which I abandoned only because I did not have enough time to investigate it deeply. So I’d like to talk about it in brief.
Recalling that I was looking for a Delaunay mesh, with cells that resemble equilateral triangles as much as possible, one thing that immediately came to my mind was the conformal map, which transforms between two spaces while preserving angles locally. Therefore, under a carefully designed conformal map, a good stippling on a uniform area is guaranteed to be transformed to a good one on a specified non-uniform area.
A simple example of a conformal map is a complex holomorphic function f, say
This transforms the following stippling points refinedPts uniformly, distributing on square [-1,1]×[-1,1] to a new points distribution transPts:
The result looks very nice in the sense of being well spaced, which can also be confirmed with its Delaunay mesh, its Voronoi mesh, and its connectivity:
So far, so good! However, this theoretically elegant approach is not easily generalizable. Finding the right f for an arbitrary target distribution (e.g. the grayscale distribution of an image) will ask for highly sophisticated mathematical skills. My numerical trials all failed, thus I’m not going to talk about the theoretical details here. If you are interested in them, there is a dedicated research field called computational conformal mapping.
Despite the elegance of the conformal mapping, I was forced back to Lloyd’s algorithm. I had to give it another thought. Yes, it doesn’t work on non-uniform cases, but it’s close! Maybe I can generalize it. After reading through some papers, I confirmed that thought. The popular method for stippling non-uniform areas indeed comes from a modification of the CVD, developed by Adrian Secord, called the weighted centroidal Voronoi diagram.
The new algorithm is similar to Lloyd’s, except that in step three, when looking for the centroid of the Voronoi cell, a variable areal density ρ(P) is considered, which is proportional to the grayscale at the location P. Thus instead of using equation 1, I can calculate the centroid according to the definition:
Clearly the integrations are much more time-consuming than in equation 1. In his paper and master thesis, Secord presents an efficient way to compute them that involves a precomputation of certain integrations. However, I noticed (without theoretical proof) that the weighted CVD can be sufficiently approximated in a much cheaper way if we accept a compromise not to stick to the exact formula of the centroid but emphasize the core idea of choosing C closer to points with larger weights.
My new idea was simple and naive. For a cell of n vertices {P_{1},…,P_{n}}, the algorithm acts as follows:
Written in the Wolfram Language, it’s this findCentroidW function:
Since the convergence of the algorithm was not obvious (and since it’s quite time-consuming), I decided to do an early stopping after dozens of iterations during the numerical experiment.
Now that everything was set up, I felt so confident in trying my functions in the real world! The first example I chose was the Lena photo.
I imported the original image, keeping both the grayscale and color versions for later use in the artistic rendering section:
For the convenience of Interpolation, I used ColorNegate. For a better visual feeling, I enhanced the edges that have large gradients:
I rescaled the image coordinates to the rectangle region [-1,1]×[-1,1] for convenience, and used Interpolation on the whole image to get a smooth description of the grayscale field:
To have a good initial point distribution, I sampled the points so that the local density is proportional to the local grayscale (though I won’t need to have this precisely, as the weighted Voronoi process will smooth the distribution anyway). By taking advantage of ContourPlot, I generated a few regions according to the level set of the densityFunc:
For each region in levelRegions, at first I sampled the points inside it on a regular grid, with the area of the grid cell inversely proportional to the level counter of the region. Then I immediately noticed the regular grid can be a steady state of my algorithm, so to ensure an isotropic result, initial randomness was needed. For that purpose, I then added a dithering effect on the points, with its strength specified by a parameter 0≤κ≤1 (where 0 gives no dithering, while 1 gives a globally random distribution):
The total number of points I had sampled as initPts was huge:
However, their quality was poor:
The weighted Voronoi relaxation process, which is similar to the one in the uniform case, was going to improve the distribution in a dramatic way, though the computation was a bit slow due to the amount of points. Notice that I forced an early stopping after 30 iterations:
In spite of only 30 iterations, the result was in my opinion fairly good (note that some visual artifacts in the following image are due to the rasterizing process during display):
I then applied the similar visual analysis on it, and found out that the pattern of the connectivity graph formed some kind of rather interesting self-similar, multi-resolution tiles:
Statistics on the interior angles of the corresponding DelaunayMesh also indicated I had indeed achieved the well-spaced distribution:
Now I felt quite satisfied!
However, as a person who likes to reinvent my own wheel, I would not stop here!
As I have mentioned before, some artistic rendering effects can be simulated based on the stippling result. Encouraged by the above result, I intended to try replicating some of them.
One effect is a simple hatching.
Inspired by the example under the Properties & Relations section in the documentation of GradientOrientationFilter, I initialized a stroke at each stippling point, with the lengths and orientations controlled by local grayscale and gradient magnitude, respectively:
There seemed to be some artifacts in the dark regions, possibly due to the naive approach, but the whole result was acceptable for me.
Another styling I tried was Pointillism.
For this simulation, I had to take care of the colors:
Then I added random deviations to the colors of each points:
I finalized the process with the morphological Opening operation and matched the histogram with the original color image using HistogramTransform:
However, I’d like to try something more “Impressionist.”
The rendering in the last example used strokes with identical size, but in practice it could often vary, with some local properties of the targets being drawn, so I tried to include that characteristic.
For the special property to be reflected, I chose the ImageSaliencyFilter (but it is, of course, totally fine to choose other ones):
Like the densityFunc, I interpolated the coarseMask and generated some level regions:
Now the stippling points could be grouped according to the coarseLevelRegions:
I then composed the stroke template as a square centered at the given point, with its color determined and randomized in a way similar to that in the last example:
Now I was ready to paint the picture layer by layer (note that the following code can be quite time consuming if you try to display coarseLayers!):
Composing the layers, I got the final Impressionist result:
Note the local vivid color blocks, especially those at the feather and hat regions, and how the rendering still preserved an accurate scene at the macro scale.
With such an accomplishment, I couldn’t wait to try the procedure on other pictures. One of them I want to share here is a photo of a beautiful seacoast taken in Shenzhen, China.
The original photo was taken on my smart phone. I chose it mainly because of the rich texture of the stones and waves. With a bit of performance optimization, I got some quite-gorgeous results in a reasonable time. (Precisely speaking, it’s 35,581 stipples finished in 557 seconds! Much faster than my bare hand!)
The hatching and Pointillism versions were also quite nice:
For now, my adventure has stopped here. However, there are still many things I’m planning to explore in the future. I may try more realistic stroke models. I may try simulating different types of the infiltration of ink on canvas. It will also surely be fun to control a robot arm with a pen drawing the graphics out in the real world on a real canvas. So many possibilities, so much stippling joy! If you love them as well, I’d like to hear suggestions from you, and hope to have some enjoyable discussions.
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For almost seven years now, millions of people have used the computational powers of Wolfram|Alpha to check their own predictions and explore fascinating topics in math, science, history, linguistics, culture, the arts, and more. Take Brian A. Carr from Jackson, Wyoming, for example. He is a classical scholar and career firefighter with Jackson Hole Fire/EMS. Carr uses Wolfram|Alpha for a variety of interests, “from helping to explain geometrical and mathematical concepts related to my study of early Greek mathematics to investigating the physics of room-and-contents fires.”
Wolfram|Alpha “was a lifesaver in my graduate work,” says Samantha Howard, who is now a physicist in the United States Air Force. Howard, who received her master’s degree in applied physics from the Air Force Institute of Technology, first discovered Wolfram|Alpha during her undergraduate years at the USAF Academy.
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Carr, Howard, and Koppel are only a few representatives of our huge community of users who are working with Wolfram|Alpha every day to explore new horizons. In honor of Mathematics Awareness Month and to encourage mathematics curiosity and exploration, we are offering 20% off subscriptions to Wolfram|Alpha Pro through April 30, 2016. Visit our website, use the promo code MATHMONTH20OFF, and start exploring!
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After four hundred years, Shakespeare’s works are still highly present in our culture. He mastered the English language as never before, and he deeply understood the emotions of the human mind.
Have you ever explored Shakespeare’s texts from the perspective of a data scientist? Wolfram technologies can provide you with new insights into the semantics and statistical analysis of Shakespeare’s plays and the social networks of their characters.
William Shakespeare (April 26, 1564 (baptized)–April 23, 1616) is considered by many to be the greatest writer of the English language. He wrote 154 sonnets, 38 plays (divided into three main groups: comedy, history, and tragedy), and 4 long narrative poems.
I will start by creating a nice WordCloud from one of his famous tragedies, Romeo and Juliet. You can achieve this with just a couple lines of Wolfram Language code.
First, you need to get the text. One possibility is to import the public-domain HTML versions of the complete works of William Shakespeare from this MIT site:
Then make a word cloud from the text, deleting common stopwords like “and” and “the”:
As you can see, DeleteStopwords does not delete all the Elizabethan stopwords like “thou,” “thee,” “thy,” “hath,” etc. But I can delete them manually with StringDelete. And with some minor extra effort, you can improve the word cloud’s style a great deal:
Now let’s analyze a tragedy more deeply. Wolfram|Alpha already offers a lot of computed data about Shakespeare’s plays. For example, if you type “Othello” as Wolfram|Alpha input, you will get the following result:
If you want to visualize the interactions among the characters of this tragedy via a social network, you can achieve this with ease using the Wolfram Language. As I did earlier with the word cloud, I need to first import the texts. In this case I want to work with all the acts and scenes from Othello separately:
Since I want to import and save the scenes for later use in the same notebook’s folder, I can do the following:
In order to create the Graph, I first need all the character names, which will be displayed as vertices. I can gather the names by noticing that each dialog line is preceded by a character name in bold, which in HTML is written like this: <b>Name</b>. Thus it is straightforward to get an ordered list containing all character names (“speakers”) from each dialog line using StringCases:
Then, using Union and Flatten, I can obtain the names of all the characters in the tragedy of Othello:
Once I have the vertices, I need to create the edges of the graph. In this case, an edge between two vertices will represent the connection between two characters that are separated by less than two lines within the dialog (similar to the Demonstration by Seth Chandler that analyzes the networks in Shakespeare’s plays). For that purpose, I will use SequenceCases to create all the edges, i.e. pairs of lines separated by less than two lines:
Before creating the graph, I need to delete the edges that are duplicated or are equivalent, like OTHELLO↔IAGO and IAGO↔OTHELLO, and the edges connecting to themselves, i.e. IAGO↔IAGO:
Finally, you can specify the size of the vertices with the VertexSize option. For example, I want the vertices’ sizes to be proportional to the number of lines per character. I can get the number of lines per character with Counts:
After this, I can use a logarithmic function to rescale the vertices to a reasonable size. I will also improve the design with VertexStyle and VertexLabels.
Since the code is getting more cumbersome, I will omit it and show only the result (for those interested in the details of the code, you can find them in the attached notebook). Also note that in the final result I’m excluding the vertex “All” since it is not a real character in the dialog:
So far, so good. Having the social network from a Shakespeare play written more than four hundred years ago is quite cool, but I’m still not 100% satisfied. I would like to visualize when these interactions occur within the dialog itself. One way to achieve this is by representing each main speaker with a different-colored bar:
Note: linesColor is a list of colors representing the lines in one scene, and linesLength is the list of the lines’ StringLength with a rescaling function. These functions involve some TextManipulation, like I did earlier to obtain the character names from the HTML version of the play. If you wish, you can see their construction in the attached notebook:
Additionally, I can mark when a particular character says a particular word—for example, the word “love” (note: the variable words is the list of words per line in the scene, created with the new function TextWords; see the attached notebook for details):
Now I can combine all of this with the social network graph and have a colorful and compact infographic about a Shakespeare tragedy:
There are so many other interesting things that I would like to explore about Shakespeare’s works and life. But I will finish with a map representing the locations at which his plays occur. I hope you got a glance of what is possible to achieve with the Wolfram Language. The only limits are our imagination:
For a few places, the Interpreter fails to find a GeoPosition, so I used Cases to obtain all the successfully interpreted locations:
Finally, I’m using Geodisk to depict geopositions by disks with a radius proportional to the number of times each location appears in Shakespeare’s plays:
Many fellow Wolfram users expressed keen interest in and came up with astonishing approaches to Shakespeare’s corpus analysis on Wolfram Community. We hope this blog will inspire you to join that collaborative effort exploring the mysteries of Shakespeare data.
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]]>This marathon is one of the six Abbott World Marathon Majors: the Tokyo, Boston, Virgin Money London, BMW Berlin, Bank of America Chicago, and TCS New York City marathons. If you are looking for things to add to your bucket list, I believe these are great candidates. Given the international appeal, let’s have a look at the runners’ nationalities and their travel paths. Our GeoGraphics functionality easily enables us to do so. Clearly many people traveled very far to participate:
The vast majority, of course, came from the US:
Let’s create a heat map to see the distribution of all US runners. As expected, most of them are from Chicago and the Midwest:
What did the race look like in Chicago? Recreating the map in the Wolfram Language, taking every runner’s running times, and utilizing my coworker’s mad programming skills, we can produce the following movie:
As you can see, the green dot is the winning runner. I am red, and the median is shown in blue. This movie made me realize that while the fastest runner was already approaching the most northern point of the course, I was still trying to meet up with my running partner! The purple bars indicate the density of runners at any given time along the race course. You might wonder what the gold curve is. That would be the center of gravity given the distribution of the runners.
The dataset also includes age division and placement within age group, gender and placement within gender group, all split times, and overall placement. The split times were taken every 5 km, at the half-marathon distance, and, of course, at the finish line. The following image illustrates the interpolated split times for all participants after deducting the starting time of the winning runner:
The graphic reflects several things about this race: runners were grouped into two waves, A and B, depending on their expected finishing time. This is illustrated by the split around 2,500 seconds at the starting line. Within each wave, runners were then grouped into corrals. Again, faster runners started in earlier corrals. Thus the later runners got started, the slower they were overall. The resulting slower split times are expressed in a much faster rise of the corresponding lines. It also took 4,503 seconds, a little over 75 minutes, for all runners to get started. In contrast, the last person crossed the finish line 19,949 seconds after the winner of this race. I was neither…
Let’s take a more detailed look at everyone’s start and finish in absolute time. We’re letting the first runner start at 0 seconds by subtracting his time from all participants’. The red dots indicate the mean of the finish time for runners with the exact same starting time:
Again, the two waves are clearly visible. The smaller breaks within each wave indicate the corral changes. But what caught my eye was the handful of people preceding the first wave. Because the dataset provides us with the names of the participants, I was able to drill down and find out whose data I was looking at: it is the “Athletes with Disabilities” (AWD), as the group is named by the Chicago Marathon administration. Checking back with the schedule of events, I was able to confirm that this group started eight minutes ahead of the first wave.
Let’s investigate a bit more and see what we can learn about this group. Of course, the very first person to cross the starting line is part of this group. Everyone else started very closely around him. We can query for the AWD subgroup by looking for everyone who started within a generous 200 seconds of the first person. We find that there were 49 members in this group:
Here is the plot of their start and finish times. It is equivalent to a zoom on the 0-second start line in the above plot:
Due to their physical disabilities, many of these runners were joined by one or two guides who helped them navigate the course. With our Nearest functionality, we can try to identify such groups. We just need to gather everyone’s time stamps, convert them to UnixTime, and define our Nearest function:
Let’s find the group of nearest people for all 49 runners by limiting the variations of their time stamps to 10 seconds over the course of the race:
Out of the 49 runners, we find that 35 ran in 15 groups of 2 or more people:
These are the groups we could identify:
I tip my hat to everyone who participated in this race. But I am in awe of people running a marathon with a physical disability. I would like to give them, as well their guides, a special shoutout!
Did I run with someone? As mentioned above, I sure did. I am lucky to have my next-door neighbor Michael as my running partner. Cursing and whining during a long run is a lot easier if you have someone on your side. Otherwise you just look crazy while mumbling to yourself. Let’s build the Nearest function:
Then we can apply it to the entire dataset. Any result of length greater than 1 indicates a running group. We find that 2,784 runners ran in 1,394 groups:
There were 1,329 groups of 2, 62 groups of 3, and 3 groups of 4. The latter were:
By the way, you will not find my and Michael’s names in any of these groups. Why? Because there was nothing in this world that could keep Michael from his tenth attempt to finish the marathon in under four hours—whereas halfway through the race I had to give in to that nagging voice telling me to take a break and walk. Just taking the first half of the race into account, here we are:
We finished only three minutes apart, but that can be a whole lot of time during a marathon. Michael came in just under four hours; I barely missed that time.
Now let’s take a look at how the race progressed split by split. The following histograms show how participants’ split times compared to the mean time at each split distance:
Interestingly, for each split the curve shows a little bump just before the 0 marker, which indicates the mean split time. To find out which runners these might be, we have to consider who the participants are. The vast majority are recreational marathon runners. We hope to stay injury free and maybe achieve a personal record, but our goal is to have a great experience and a rush of endorphins. We are not there to win and collect prize money. But, as Michael did above, one thing that people might attempt is to break the illusive four-hour mark. To beat four hours, a runner—let’s call her “Molly”—has to average 341.517 s/km, or 9 minutes and 9 seconds per mile:
To make sure Molly comes in under four hours, let’s assume she runs at a pace five seconds faster per kilometer, 336.517 s/km. By not allowing any change of pace, we are basically turning Molly into a robot. But let’s see where her split times (indicated in red) fall compared to the mean at each of the kilometer markers. Indeed, Molly’s split times match the “hump,” and thus are a representation of all runners trying to finish the marathon in less than four hours:
As can be seen in the above histograms, with each split we plot more bins representing fewer runners, while the variations from the mean steadily increase. Here is another look at the same fact, just from a different angle. Again taking the differences of the runners’ split times to the mean, and then sorting them from smallest to largest, we can see how the differences between the fastest and slowest runners steadily increase over the course of the race:
Again, the group of people trying to finish in under four hours is nicely visible in the small hump to the left of the y axis. How many people did make it in under four hours? We could not make this number up: it was exactly 11,111 people, or 29.7% of all participants:
As mentioned above, I could not keep pace with Michael after about halfway through the race. But let’s look at “keeping pace” and how consistent people ran their race. The dataset provides all the information we need to look at everyone’s average pace and absolute variations from it at each split. Adding up those variations per person gives us the following picture:
The maximum of accumulated variations from the average pace is around 10 minutes. I averaged 9 minutes and 16 seconds:
My variations from that average added up to almost three minutes:
In the charts below, we are looking at the distribution of those variations versus a runner’s finishing time. Since a slower runner takes more time between splits and thus automatically accumulates more minutes and more variations, we additionally normalized the pace variation by the corresponding finishing time:
Of course, these pace variations cause people to pass each other. Let’s have a quick look at how often this happened. We counted an amazing 276,121,258 occurrences of runners’ position changes. Below is an illustration. Inside the attached notebook, please hover over the data points to see the number of takeovers at a given distance:
To explain the numerous peaks, we should have another look at the race. Every mile or two, aid stations were providing runners with fluids, medical assistance, and other necessities. These aid stations were about two city blocks long, giving runners plenty of opportunities to move through and to avoid crowds. Consider the aid stations on the map:
Also consider their locations along the course by using our new GeoDistanceList function:
We can nicely match the peaks with the locations of the aid stations. At each of these points, a huge number of runners change their paces, resulting in the jump in takeovers. While taking in fluids, one runner might choose to walk while another just slows down but continues to run. A third runner might not utilize the station at all and run through it. Turns out I am not very gifted when it comes to drinking while running, so I walk whenever necessary.
Interestingly, a Histogram3D of time versus distance versus the number of takeovers looks like the city of Chicago itself:
Running a marathon does not just take a good number of months of training, battles with injuries, and bouts of laziness (as well as a good sense of the craziness of this endeavor). It also takes a financial commitment. Race registration and travel costs can add up to an intimidating sum of money. This made me wonder if there is a correlation between travel distance and finishing time, i.e. can I assume that the farther you have to travel and the more money you have to spend on the event, the better you are as a runner? The following plot shows the finishing time versus travel distance to the US. Upon hovering inside the notebook, you can see the runners’ countries, their finishing times, and their overall placement in the race:
Clearly my assumption is incorrect. We do see a small number of runners from Kenya and Ethiopia who traveled thousands of miles and came in first. But we also see runners who traveled all the way from India, New Zealand, Indonesia, Swaziland, and Singapore who finished in more than six or seven hours. The means for these countries are all around six hours.
Let’s see if another assumption can be proven wrong, e.g. if the travel expense is not as prohibitive as thought, does the number of runners from a country decrease with increasing travel distance? And could it be true that the more runners a country has in the race, the higher its GDP per capita is? In the notebook, hover over each data point in the charts below to see the country, number of runners from that country, and travel distance or GDP per capita:
The data is not as obvious as one might think. More than 28,000 participants came from the US, whereas only a single person came from countries such as Réunion and Mauritius. We do have a number of countries with less wealth and only single-runner representation. But the single-runner representation also holds true for Qatar and Luxembourg—both known for their financial muscle.
I’ll admit that the country of origin might not be as much of a statement about the size of one’s wallet or someone’s performance as I might have thought. What about age?
Marathons seem to appeal mainly to people in their mid-twenties to mid-forties. And, of course, the higher your age, the better your chances of winning your division. But what is interesting to see is that this is not actually a sport favoring the younger athletes. The fastest times were achieved by the 40–44 age division. So I might still have my Olympic years ahead of me!
To add a note of obscurity: have you ever considered if your name is any indication of your performance? Or if there are other runners by your name in this exact race? There are many shared first and last names. If you were a “Cabada” or a “Zac” in this race, you did awfully well:
You may have guessed the most common first name: there were 641 Michaels. The leading last name was, also not very surprising, “Smith” with a count of 157. Of course, these numbers decrease considerably when we look at shared full names:
And the most common full names and their counts are:
The combination of my family watching on the sidelines, including my mother visiting from Germany, the outstanding work of all the volunteers, and the huge crowds of spectators and the entertainment they provided, all made for a memorable race. Plus the weather, which is usually a liability in Illinois, was just impeccable. Both Michael and I had a blast, which I think is visible using ImageCollage:
But as it turns out, not just the event itself was fun. This was a great dataset for me to play around with and learn a lot more about the capabilities of the Wolfram Language. I am not a developer, but I greatly enjoyed this opportunity to combine my professional and personal lives. If you are interested in more scientific approaches to the topic of marathon running, you might find this article and this article intriguing.
But most importantly, registration is now open for the 2016 event!
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Handbook of Mathematics, sixth edition
This guidebook to mathematics by I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muhlig contains a fundamental working knowledge of mathematics, which is needed as an everyday guide for working scientists and engineers, as well as for students. This newer edition emphasizes those fields of mathematics that have become more important for the formulation and modeling of technical and natural processes, namely numerical mathematics, probability theory and statistics, and information processing. Besides many enhancements and new paragraphs, new sections on geometric and coordinate transformations, quaternions and applications, and Lie groups and Lie algebras have also been included.
Advanced Calculus Using Mathematica: Notebook Edition
Keith Stroyan’s latest work is a complete text on calculus of several variables written in Mathematica notebooks. The eText has large, movable figures and interactive programs to illustrate things like “zooming in” to see “local linearity.” In addition to lots of traditional-style exercises, the eText also has sections on computing with Mathematica. Solutions to many exercises are in closed cells of the eText.
Handbook of Linear Partial Differential Equations for Engineers and Scientists, second edition
Including nearly 4,000 linear partial differential equations (PDEs) and a database of test problems for numerical and approximate analytical methods for solving linear PDEs and systems of coupled PDEs, Andrei D. Polyanin and Vladimir E. Nazaikinskii have created a comprehensive second edition of their handbook. The book also covers solutions to numerous problems relevant to heat and mass transfer, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrodynamics, diffraction theory, quantum mechanics, chemical engineering sciences, electrical engineering, and other fields.
Mathematical Science of the Developmental Process (Japanese)
This book by Takashi Miura uses Mathematica to introduce and explore the developmental process, in which a simple, spherical, fertilized egg becomes a complex adult structure. The process is very difficult to understand, and the mechanism behind it has yet to be elucidated. Since no fundamental equation of this process has been established, we need prototyping processes, which means quick formulation of simple phenomenological models and verification by simulation and analysis.
Single Variable Calculus with Early Transcendentals
A comprehensive, mathematically rigorous exposition, this text from Paul Sisson and Tibor Szarvas blends precision and depth with a conversational tone to include the reader in developing the ideas and intuition of calculus. A consistent focus on historical context, theoretical discovery, and extensive exercise sets provide insight into the many applications and inherent beauty of the subject.
If you are not a programmer but you need to analyze data, Sergiy Suchok’s new book will show you how to use Mathematica to take just a few strings of intelligible code to solve huge tasks, from statistical issues to pattern recognition. If you’re a programmer, you will learn how to use the library of algorithms implemented in Mathematica in your programs, as well as how to write algorithm testing procedures. Along with intuitive queries for data processing and using functions for time series analysis, we will highlight the nuances and features of Mathematica, allowing you to build effective analysis systems.
Mathematica: A Problem-Centered Approach, second edition
The second edition of Roozbeh Hazrat’s textbook introduces the vast array of features and powerful mathematical functions of Mathematica using a multitude of clearly presented examples and worked-out problems. Based on a computer algebra course taught to undergraduate students of mathematics, science, engineering, and finance, the book also includes chapters on calculus and solving equations, as well as graphics, thus covering all the basic topics in Mathematica. With its strong focus on programming and problem solving, and an emphasis on using numerical problems that do not require any particular background in mathematics, this book is also ideal for self-study and as an introduction for researchers who wish to use Mathematica as a computational tool.
Estructuras Discretas con Mathematica (Spanish)
This book by Enrique Vilchez Quesada provides a theoretical and practical overview for students studying discrete structures within the curriculum of computer engineering and computer science. The major contribution of this work, compared to other classical textbooks on this subject, consists of providing practical solutions to real-world problems in the context of computer science by creating different examples and solutions (programs, in most cases) using the renowned commercial software Mathematica.
Looking for more Wolfram technologies books? Don’t forget to visit Wolfram Books to browse by both topic and language!
]]>In 1828, an English corn miller named George Green published a paper in which he developed mathematical methods for solving problems in electricity and magnetism. Green had received very little formal education, yet his paper introduced several profound concepts that are now taught in courses on advanced calculus, physics, and engineering. My aim in writing this post is to give a brief biography of this great genius and provide an introduction to GreenFunction, which implements one of his pioneering ideas in Version 10.4 of the Wolfram Language.
George Green was born on July 14, 1793, the only son of a Nottingham baker. His father noticed young George’s keen interest in mathematics, and sent him to a local school run by Robert Goodacre, a well-known science popularizer. George studied at Goodacre Academy between the ages of eight and nine, and then went to work in his father’s bakery. Later he ran a corn mill built by his father in Sneinton, near Nottingham. He is said to have hated his work at the bakery and the corn mill, and regarded it as annoying and tedious. In spite of his onerous duties, George appears to have continued studying mathematics in his spare time, retreating to the top floor of the 16-meter-high mill, shown above, for this purpose. In 1828, he published the results of his rigorous self-study in “An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism,” one of the most influential mathematical papers of all time.
Green’s paper of 1828 introduced the potential function, which is well known to students of physics. He also proved a form of Green’s theorem from advanced calculus in this paper. Finally, he introduced the notion of a Green’s function that, in one form or another, is familiar to students of engineering, and is the theme for this post. By sheer chance, Sir Edward Bromhead, a founder of the Analytical Society, purchased and read a copy of Green’s paper. With his encouragement, Green entered Gonville and Caius College in Cambridge University at the age of forty, and eventually became a fellow of the college. He continued to publish papers until his untimely death in 1841, possibly due to lung complications arising from his work at the corn mill. Sadly, recognition for his mathematical work had to wait until 1993, when a plaque was dedicated to his memory in Westminster Abbey. Today, the Green’s Mill and Science Centre in Nottingham carries on the work of promoting George Green’s reputation as one of the greatest scientists of his age.
I will now give an introduction to GreenFunction using concrete examples from electrical circuits, ordinary differential equations, and partial differential equations.
The basic principle underlying a Green’s function is that, in order to understand the response of a system to arbitrary external forces, it is sufficient to understand the system’s response to an impulsive force of the DiracDelta type.
As an illustration of the above principle, consider a circuit that is composed of a resistor R and an inductor L, and is driven by a time-dependent voltage v[t], as shown below:
The current i[t] in the circuit can then be computed by solving the differential equation:
L i´(t)+R i(t)==v(t)
Let’s assume that the voltage source is a battery supplying a unit voltage. Next, suppose that you close the switch S for a fleeting moment at time t = s and then quickly throw it open again. The current induced in the circuit by this impulsive action can be computed by applying GreenFunction to the left-hand side of the above differential equation:
The initial value of the current is assumed to be zero, since the switch was open until time t = s:
Here is the result given by GreenFunction for this example:
The following plot for s = 1 shows that the current is 0 for all times t < 1, then rises instantaneously to its peak value at t = 1, and finally decreases to 0 with the passage of time:
The behavior of the circuit in the above situation is usually called its impulse response, since it represents the response of the circuit to an impulsive voltage.
Next, suppose that you close the switch at time t = 0 and leave it closed at all later times. Thus the voltage steps up from its initial value 0 to a constant value 1, and can be modeled using the HeavisideTheta function:
The step voltage can be visualized as follows:
You can now compute the current in the circuit by performing the following integral involving the voltage and the Green’s function:
The integral computed above is essentially a weighted sum of the Green’s function with the voltage source at all times s prior to a given time t, and is called a convolution integral.
As the plot below shows, the current for the step voltage source gradually increases from its value 0 at t = 0 to a steady-state value:
The behavior of the circuit in the above situation is usually called its step response, since it represents the response of the circuit to a step voltage.
Finally, suppose that the voltage source supplies an alternating voltage—for example:
You can once again compute the current in the circuit by performing a convolution integral of the voltage with the Green’s function, as shown below:
You can also obtain the result using DSolveValue as follows:
As the plot below shows, the current settles down to a steady alternating pattern for large values of the time:
To summarize, the Green’s function encodes all the information that is required to study the response of the circuit to any external voltage. This magical property makes it an indispensable tool for studying a wide variety of physical systems.
The two-step procedure for solving the differential equation associated with a circuit, which I discussed above, can be applied to any linear ordinary differential equation (ODE) with a forcing term on its right-hand side and homogeneous (zero) initial or boundary conditions. For example, suppose you wish to solve the following second-order differential equation:
Assume that the forcing term is given by:
Also, suppose that you are given homogeneous boundary conditions on the interval [0,1]:
As a first step in solving the problem, you compute the Green’s function for the corresponding differential operator (left-hand side) of the equation:
The following plot shows the Green’s function for different values of y lying between 0 and 1. Each instance of the function satisfies the zero boundary conditions at both ends of the interval:
You can now compute the solution of the original differential equation with the given forcing term using a convolution integral on the interval [0,1], as shown below:
Here is a plot of the solution, which shows that it satisfies the homogeneous boundary conditions for different values of the parameter a:
Green’s functions also play an important role in the study of partial differential equations (PDEs). For example, consider the wave equation that describes the propagation of signals with finite speed, and that I discussed in an earlier post. In order to compute the Green’s function for this equation in one spatial dimension, use the wave operator (left-hand side of the wave equation), which is given by:
Here, x denotes the spatial coordinate that ranges over (-∞,∞), t denotes the time that always ranges over [0,∞), and u[x,t] gives the displacement of the wave at any position and time.
You can now find the Green’s function for the wave operator as follows:
The following plot of the Green’s function shows that it becomes 0 outside a certain triangular region in the x-t plane, for any choice of y and s (I have chosen both these values to be 0). This behavior is consistent with the fact that the wave propagates with a finite speed, and hence signals sent at any time can only influence a limited region of space at any later time:
The Green’s function obtained above can be used to solve the wave equation with any forcing term, assuming that the initial displacement and velocity of the wave are both zero. For example, suppose that the forcing term is given by:
You can solve the wave equation with this forcing term by evaluating the convolution integral
The following plot shows the standing wave generated by the solution:
Finally, I note that the same solution can be obtained by using DSolveValue with homogeneous initial conditions, as shown below:
Green’s functions of the above type are called fundamental solutions and play an important role in the modern theory of linear partial differential equations. In fact, they provided the motivation for the theory of distributions that was developed by Laurent Schwartz in the late 1940s.
The ideas put forward by George Green in his paper of 1828 are stunning in their depth and simplicity, and reveal a first-rate mind that was far ahead of the times during which he lived. I have found it very inspiring to study the life and work of this great mathematician while implementing GreenFunction for Version 10.4 of the Wolfram Language.
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