August 30, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

Back in 325 BC, Aristotle talked about which polyhedra can fill space, and noted that regular tetrahedra could fill space.

Around 1470 AD, Regiomontanus showed that Aristotle was wrong. He also found the spot where a statue on a pedestal appears the largest, as shown in the Demonstration “The Statue of Regiomontanus”.

In 1896, Minkowski tried to solve the problem of how well tetrahedra could pack. He failed. But he did introduce many valuable tools to math, such as “The Minkowski Sum of Two Triangles”.

In 1900, Hilbert tried the problem of tetrahedra packing and included it as a part of problem 18 in his list of unsolved problems. Hilbert is also famous for the Hilbert curve and “The Hilbert Hotel”.

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July 13, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

Is it possible to have a pair of nonstandard dice with the same odds as regular dice?

Sure. You just need to know how to calculate the odds, and how to determine what different numbers could be on the faces to give the same odds. Let’s start with some tables.

The addition table is one of the first tables learned in school. Here is
one way to present an addition table in Mathematica.

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June 30, 2010 — Wolfram Blog Team

Bruce Torrence, PhD, chair of the Department of Mathematics at Randolph-Macon College, says he’s engaging his students in mathematics more than ever before thanks to a single Mathematica command. That command is Manipulate.

Professor Torrence calls the ability to create instant dynamic interfaces a “real game changer” for helping students understand mathematics. He says, “Once you play with a Manipulate and interact with the sliders and buttons, you really develop your intuition as to how the underlying mechanisms are interacting and working.”

In this video, Professor Torrence shares an example of how he used Mathematica to turn a previously tedious lesson into a highly compelling, interactive classroom activity.

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June 16, 2010 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

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.

Code to generate a table
The F7 Farey sequence and its associated circular bends for Ford Circles

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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:

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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.

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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:

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September 9, 2009 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

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.

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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.

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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:

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