April 23, 2010 — Carol Cronin, Public & Community Relations
April is Mathematics Awareness Month, and this year’s theme is “Mathematics and Sports.” It’s sponsored by the Joint Policy Board for Mathematics to promote the importance of math, and schools and organizations nationwide are participating by hosting presentations, competitions, and poster contests for students from elementary school through graduate school.
Wolfram Research is proud to support Mathematics Awareness Month again this year. To remind students everywhere that math can be fun, we have provided complimentary Mathematica for Students licenses to several competitions this month to be distributed as prizes, including these:
April 8, 2010 — Wolfram Blog Team
Thousands of universities around the world take advantage of Mathematica‘s revolutionary developments for engineering, science, economics, mathematics, and more, for a vast number of courses across campus.
One of those schools is Truman State University.
Dana Vazzana, an associate professor of mathematics at Truman, integrates Mathematica into every course she teaches. She says using Mathematica with her students creates a dynamic classroom where students gain deeper understanding of concepts and richer insights into real-world applications of mathematics. “Anything that gets them that involved and that excited and makes them want to go and work some more has just got to be a good thing,” explains Professor Vazzana.
October 28, 2009 — Oleksandr Pavlyk, Kernel Technology
The “Problems and Solutions” section of The American Mathematical Monthly journal has always been a source of interesting problems to keep me entertained. Their solutions often require ingenuity. The problems in the October issue were no exception.
I always analyze and explore these problems in Mathematica. Being a kernel developer, I see whether Mathematica can indeed find a solution. This last issue has challenging problems, and it was particularly gratifying to observe that Mathematica could solve them right out of the box. So here are my solutions to three of the paraphrased problems:
September 9, 2009 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
Number 9, number 9, number 9.
When a number has a lot of nines in it, like .99999999999999999, many computer systems can run into rounding problems. Fortunately, Mathematica can handle both exact and numeric forms. Here are exact forms of various
numbers whose numeric forms have lots of nines.
January 26, 2009 — Yu-Sung Chang, Technical Communication & Strategy
One of the areas I contributed to Mathematica 7 was support for splines. The word “spline” originated from the term used by ship builders referring to thin wood pieces.
Over the last 40 years, splines have become very popular in computer graphics, computer animation, and computer-aided design fields. From containers for household goods to state-of-the-art airplanes, it is hard to find any industrial product without spline surfaces. Also, they are widely used in other mathematical studies, such as interpolation and approximation.
Through its integration of numerics, symbolics, and graphics, Mathematica has the opportunity to go much further with splines than has ever been possible before. Mathematica has had basic spline packages for a long time. But in Mathematica 7 we decided to make highly general spline support a core feature of the system.
Splines give another way to represent classes of functions. For decades, mathematicians had been using polynomials for numerical analysis. In early 20th century, with advances in approximation theory, spline functions were beginning to emerge. The basic idea is simple. In essence, they consist of piecewise polynomials with local supports.
Since Version 5.1, Mathematica has offered general support for piecewise functions, both numerically and symbolically. In Mathematica 7, the B-spline functions can be expanded using PiecewiseExpand. For example, a uniform cubic B-spline basis function can be expanded to the following.
December 18, 2008 — Roger Germundsson, Director of Research & Development
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:
August 7, 2008 — Chris Boucher, Consultant, Special Projects Group
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, not 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):
May 20, 2008 — Ulises Cervantes-Pimentel, Senior Kernel Developer
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.
April 21, 2008 — Todd Rowland, Academic Director, Wolfram Science Summer School
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:
|Zonohedron Turned Inside Out
|Particle Moving around
Two Extreme Black Holes
March 13, 2008 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
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 14th, 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 on any of them to reach an interactive math demonstration. Enjoy!