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Education & Academic

The Circles of Descartes

Somewhere, you've likely been forced to learn how fractions work, and how to calculate 2/7 + 2/5. To some extent, fractions have been falling out of favor in the world, losing out to decimals. The New York Stock Exchange gave up fractions on April 9, 2001. Much of the time, a decimal is okay. Sometimes, though, especially in mathematics, exact values are desired. Instead of a value being 3.00000000...00727..., it is exactly 3. Or exactly 10/35 + 14/35 = 24/35. For fractions themselves, the Farey sequence is quite interesting—the reduced fractions between 0 and 1 where the denominator is less than or equal to a particular value, like 7. For example, the F7 Farey sequence is the the first row in the following block. The next row has the denominator. The third row is twice the reciprocal of the denominator squared. The fourth row is the denominator from the third row.
Ed Pegg Jr, Scientific Information Editor, Scientific Information Group
Education & Academic

Get Your Game On for Mathematics Awareness Month!

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:
Carol Cronin, Sponsorship Coordinator, Partnerships
Education & Academic

Give Your Classroom an Edge with Mathematica

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.
Wolfram Blog Team
Education & Academic

Mathematica and The American Mathematical Monthly’s “Problems and Solutions” Section

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:
Oleksandr Pavlyk, Senior Kernel Developer, Algorithms R&D
Education & Academic

9–9–9

Number 9, number 9, number 9. The Beatles' "Revolution 9" has the above loop, and their version of Rock Band is being released today. The movie 9 comes out today, too. 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.
Ed Pegg Jr, Scientific Information Editor, Scientific Information Group
Education & Academic

Splines Come to Mathematica

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 the 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.
Yu-Sung Chang, Technical Communication & Strategy Group
Education & Academic

Mathematica 7, Johannes Kepler and Transcendental Roots

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:
Roger Germundsson, Director of Research & Development
Education & Academic

Seeing beyond a Theorem

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):
Chris Boucher, Special Projects Group
Education & Academic

Differential Geometry Carved in Stone

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.

Ulises Cervantes-Pimentel, Senior Kernel Developer
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