News, Views & Insights
Today we celebrate the arrival of the 36th (x.x) version of the Wolfram Language and Mathematica: Version 14.1. We’ve been doing this since 1986: continually inventing new ideas and implementing them in our larger and larger tower of technology. And it’s always very satisfying to be able to deliver our latest achievements to the world.
Today we celebrate a new waypoint on our journey of nearly four decades with the release of Version 14.0 of Wolfram Language and Mathematica. Over the two years since we released Version 13.0 we’ve been steadily delivering the fruits of our research and development in .1 releases every six months. Today we’re aggregating these—and more—into Version 14.0.
In the middle of last year, we finished our decade-long project to reinvent Mathematica, and we released Mathematica 6. We introduced a great many highly visible innovations in Mathematica 6—like dynamic interactivity and computable data. But we were also building a quite unprecedented platform for developing software. And even long before Mathematica 6 was released, we […]
Calculus II is one of my favorite classes to teach, and the course I’ve probably taught more than any other. One reason for its special place in my heart is that it begins on the first day of class with a straightforward, easily stated, yet mathematically rich question: what is the area of a curved region? Triangles and rectangles—figures with straight sides—have simple area formulas whose derivation is clear. More complicated polygons can be carved up into pieces that are triangles and rectangles. But how does one go about finding the area of a blob?
After simplifying the blob to be a rectangle whose top side has been replaced with a curve, the stage is set for one of the classic constructions in calculus. The area of our simplified blob, reinterpreted as the area under the graph of a function is approximated using a series of rectangles. The approximation is obtained by partitioning the x axis, thus slicing the region into narrow strips, then approximating each strip with a rectangle and summing all the resulting approximations to produce a Riemann sum. Taking a limit of this process by using more and narrower rectangles produces the Riemann integral that forms the centerpiece of Calculus II. Several Demonstrations from the Wolfram Demonstrations Project, including "Riemann Sums" by Ed Pegg Jr, "Common Methods of Estimating the Area under a Curve" by Scott Liao and "Riemann Sums: A Simple Illustration" by Phil Ramsden show that this construction and images like the one below from "Riemann Sums" are part of the iconography of calculus.The New York Times recently published an “Op-Chart” by Tommy McCall on its Opinion page showing what your returns would have been had you started with $10,000 in 1929 and invested it in the stock market, but only during the administrations of either Democratic or Republican presidents. His calculations showed that if you had invested only during Republican administrations you would now have $11,733 while if you had invested only during Democratic administrations you would now have $300,671. Twenty-five times as much!
That’s a pretty dramatic difference, but does it stand up to a closer look? Is it even mathematically plausible that you could have essentially no return on your investment at all over nearly 80 years, just by choosing to invest only during Republican administrations?
To answer these questions, I of course turned to Mathematica.
And the answer is that yes, it is mathematically plausible, using the assumptions made by McCall. My analysis using historical Dow Jones Industrial Average data available in Mathematica’s FinancialData function roughly matches the figures in the Times, which used Standard & Poor’s data. (I used the Dow because it’s more convenient, not because I think it’s a better measure.)
But the fact that they are correct doesn’t mean the figures are even remotely meaningful. Here are some problems with the New York Times’ Op-Chart:A few times a year they would arrive. Email dispatches from an adventurous explorer in the world of geometry. Sometimes with subject lines like “Phenomenal discoveries!!!” Usually with images attached. And stories of how Russell Towle had just used Mathematica to discover yet another strange and wonderful geometrical object. Then, this August, another email arrived, […]
The 2008 United States presidential election is arguably the most interesting US presidential election in my lifetime.
Already, millions of Americans have registered to vote for the first time in their lives.
Regardless of the outcome, America is going to elect either its first black president or its first female vice president.
America will elect a sitting US senator to the highest office in the land—which, until now, has only occurred twice in US history (Warren Harding and John F. Kennedy were US senators).
Both presidential candidates were born outside of the continental United States.
If elected, John McCain will be the oldest sitting US president upon ascension to the presidency.
Never before in US history has there been such a large disparity in age between the two presidential candidates, either.
It’s also the first election you can analyze using Mathematica 6.Mathematics is a notoriously technical subject that prizes exactingly precise statements. The square of the hypotenuse of a right triangle is the sum of the squares of the legs, not the sum of their cubes, nor the difference of their squares. Such precision produces the clarity that makes the subject so powerful, but occasionally it comes at the cost of easy understanding. Indeed, more-complicated mathematical statements often sound bewildering upon first reading. Take the following theorem in plane geometry (deep breath...):
Let ABC be a triangle. Let DEF be parallel to AC with D on AB and E on BC. Let FGH be parallel to AB with G on BC and H on AC. Let , , and be the radii of the incircles , , and of the triangles ABC, DBE, EFG and HGC, respectively. If F is outside of ABC, then . Got it? Many theorems of mathematics, including this one, are easier to communicate by picture than by words. Here’s the scenario described in the theorem (images in this post are produced by slightly modified versions of the code for the Demonstration “The Radii of Four Incircles,” which is one of nearly 200 Demonstrations about theorems in plane geometry written by Jay Warendorff for the Wolfram Demonstrations Project):