Everyone who has been through high-school mathematics knows about polynomial equations. But what about equations involving other functions? Say equations like x == 1 - Sin[x]. These are transcendental equations, and they show up in a zillion different mathematical application areas. But they’re rarely talked about—perhaps because in some sense they’ve been an embarrassment: mathematics has had very little to say about them. Polynomial equations and the algebraic numbers that represent their solutions have been one of the great success stories of pure mathematics. Over the past half millennium, a huge mathematical structure has been built on polynomial equations. But almost nothing has been done with transcendental equations. It’s not that they’re not important. In fact, what many people consider the very first computer—made of wood by Wilhelm Schickard in 1623—was built specifically to help in getting solutions to equations of the form x == 1 - e Sin[x]. Johannes Kepler was in the process of constructing his Rudolphine astronomical tables—and his killer technology for finding the longitude of a planet at a given time required solving what’s now called Kepler’s equation: essentially the transcendental equation x == 1 - e Sin[x]. With considerable effort, and probably computer support, Kepler made a table of solutions to this equation:

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 r, r1, r2 and r3 be the radii of the incircles O, O1, O2 and O3 of the triangles ABC, DBE, EFG and HGC, respectively. If F is outside of ABC, then r = r1 + r2 + r3. 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):

I work on geometric computation and graphics in Mathematica, and for Mathematica 6 I was responsible for our new surface-drawing capabilities. When I talk about my work at university mathematics departments, I am often told that I just have to see what the department has tucked away in some corner of its building: plaster casts of intriguing mathematical surfaces, created in the early part of the twentieth century to illustrate the achievements of the field of differential geometry. It’s been very difficult even to reproduce those plaster casts, let alone to go beyond them—each one represents a sophisticated combination of symbolic mathematics, numerics and geometry. But with Mathematica, we now have just the combination of capabilities that are needed. And I always find it fun to reproduce those plaster-cast surfaces—often with single lines of Mathematica code, usually centered on the function ParametricPlot3D. With 3D printing, I’ve even been able to make my own physical versions of lots of these surfaces.

On Sunday, April 13, 2008, John Wheeler passed away at the age of 96. He was a central figure in twentieth-century physics, in the middle of it all, working on the H-bomb and studying black holes. His legacy in physics is continued in his influence on a vast number of students, and their students in turn. His contributions were many. Some have found their way into Demonstrations:

Gravitation versus Curved Spacetime

Zonohedron Turned Inside Out

Particle Moving around Two Extreme Black Holes

Schwarzschild Space-Time Embedding Diagram

But I want to mention his role in the history of computation in science. In a sense, he is a spiritual grandfather of NKS and of Mathematica.

Pi (π, the ratio of the circumference of a circle to its diameter), its older brother the golden ratio phi (φ) and the much younger e and i are the most famous numbers in mathematics. Pi is everywhere: not only in circles and spheres, but also in the results of all kinds of integrals, sums and products, as well as in number theory and physics. The personality of π is largely unknown: irrational, transcendental, possibly and probably normal. Because of π’s importance, its digits (3.14159265...) have an almost cult following. The first few digits, 3.14, correspond to notation for March 14, which was first celebrated as Pi Day in 1988, in the San Francisco Exploratorium. Wolfram Research has the most π presence on the web, with material at the Wolfram Functions Site (pi page, pi visualizations), MathWorld (pi, circle, sphere) and the Wolfram Demonstrations Project (pi, circle, sphere, disk, wheel), not to mention several built-in Mathematica symbols (Pi, EllipticPi, PrimePi). For NUMB3RS episode 314 (“Takeout”), we helped to fold many hidden π references into the script review and math notes. The writers, director, cast and crew added many more. The opening Black Box, for example: a 3-course meal, 1 restaurant, 4 robberies, 1592 death squad murders. Charlie mentions a circle-circle tangency joke not working, right before a James Bond reference (007---circle, circle, tangent). Below are a few of our π-related Demonstrations. Click any of them to reach an interactive math demonstration. Enjoy!

While MathWorld continues to be the most popular and most visited mathematics site on the internet, and while its mathematical content continues to steadily grow and expand, MathWorld readers will today notice more immediate visual changes. Design changes and major new pieces of functionality are generally years in the making for large informational websites like MathWorld. The last time the site received a major infrastructure upgrade was in July of 2005 (see “MathWorld Introduces New Interactive Features for Teachers and Students,” MathWorld headline news, July 6, 2005). On February 8, we introduced a major update of the MathWorld site featuring improved navigation, higher-quality typesetting and links to interactive Demonstrations. The new features introduced on MathWorld include: New streamlined “platformed” look and feel New interactive Demonstration collections and links Improved mathematical typesetting Collapsible navigation link trails More-prominent ways to contribute to MathWorld Each of these elements is described in more detail below.

Most calculus students might think that if one could compute indefinite integrals, it would always be easy to compute definite ones. After all, they might think, the fundamental theorem of calculus says that one just has to subtract the values of the indefinite integral at the end points to get the definite integral.

So how come inside Mathematica there are thousands of pages of code devoted to working out definite integrals---beyond just subtracting indefinite ones? The answer, as is often the case, is that in the real world of mathematical computation, things are more complicated than one learns in basic mathematics courses. And to get the correct answer one needs to be considerably more sophisticated. In a simple case, subtracting indefinite integrals works just fine. Consider computing the area under a sine curve, which equals

I work on computational algorithms for Mathematica, and I always like to see that what I do is helpful in solving real-world problems. When I heard about the I-35W bridge collapse, I wanted to see if anything could be learned from computing the mechanics of the bridge with Mathematica. Large packages have been written for doing structural computations with Mathematica. But I wanted to start from first principles to try to understand the whole picture. A truss bridge can be thought of as a graph, with trusses as edges and joints as nodes, as in the picture here: