November 6, 2013 — Wolfram Blog Team
Last month, students in the midterm review session of Harvard’s Math 21a class received a lesson in Mathematica they would not soon forget. Professor Oliver Knill coded a 3D-animated Miley Cyrus swinging on a wrecking ball to the beat of her song (by the same name). Knill used the same principles of mathematics that his class was reviewing for the midterm—and now he just may be the coolest professor ever.
October 24, 2013 — Wolfram Blog Team
By now, most of you students are likely getting into the thick of the academic year, preparing for the first wave of exams and projects and presentations to come your way… But don’t freak out just yet! Here’s a list of Wolfram’s most recent apps and programs that might help make your life a little easier. After all, it never hurts to have a few powerful resources on your side.
September 30, 2013 — Johan Rhodin, Kernel Developer
What is the cost of extending a warranty for a car? I’d be interested to know, since my car broke down just past the 100,000 mile marker on a road trip through America. With Mathematica 9 comes complete functionality for reliability analysis that can help us analyze systems like cars. I thought it might be worthwhile to take Mathematica for a spin and look at how some technical systems can be modeled and analyzed.
September 19, 2013 — Itai Seggev, Mathematica Algorithm R&D
I love Maxwell’s equations. As a freshman in college, while pondering whether to major in physics, computer science, or music, it was the beauty of these equations and the physical predictions that can be elegantly extracted from them that made me decide in favor of physics. On a more universal level, the hints in Maxwell’s equations led Einstein to write Zur Elektrodynamik bewegter Körper (“On the Electrodynamics of Moving Bodies”), more commonly known as Einstein’s first paper on the theory of relativity. The quantum version of the equations, quantum electrodynamics (QED), remains our most successful physical theory, with predictions verified to 12 decimal places. There are many reasons to love Maxwell’s equations. And with Mathematica 9′s new vector analysis functionality, exploring them has never been easier.
August 27, 2013 — Michael Trott, Chief Scientist
(This is the third post in a three-part series about electrostatic and magnetostatic problems involving sharp edges.)
In the first blog post of this series, we looked at magnetic field configurations of piecewise straight wires. In the second post, we discussed charged cubes and orbits of test particles in their electric field. Today we will look at magnetic systems, concretely, mainly at a rectangular bar magnet with uniform magnetization.
June 26, 2013 — Jon McLoone, International Business & Strategic Development
My government (I’m in the UK) recently said that children here should learn up to their 12 times table by the age of 9. Now, I always believed that the reason why I learned my 12 times table was because of the money system that the UK used to have—12 pennies in a shilling. Since that madness ended with decimalization the year after I was born, by the late 1970s when I had to learn my 12 times table, it already seemed to be an anachronistic waste of time.
June 3, 2013 — Oleksandr Pavlyk, Kernel Technology
I am a junkie for a good math problem. Over the weekend, I encountered such a good problem on a favorite subject of mine–probability. It’s the last problem from the article “A Mathematical Trivium” by V. I. Arnol’d, Russian Mathematical Surveys 46(1), 1991, 271–278.
It’s short enough to reproduce in its entirety: “Find the mathematical expectation of the area of the projection of a cube with edge of length 1 onto a plane with an isotropically distributed random direction of projection.” In other words, what is the average area of a cube’s shadow over all possible orientations?
This blog post explores the use of Mathematica to understand and ultimately solve the problem. It recreates how I approached the problem.
A century ago, Srinivasa Ramanujan and G. H. Hardy started a famous correspondence about mathematics so amazing that Hardy described it as “scarcely possible to believe.” On May 1, 1913, Ramanujan was given a permanent position at the University of Cambridge. Five years and a day later, he became a Fellow of the Royal Society, then the most prestigious scientific group in the world at that time. In 1919 Ramanujan was deathly ill while on a long ride back to India, from February 27 to March 13 on the steamship Nagoya. All he had was a pen and pad of paper (no Mathematica at that time), and he wanted to write down his equations before he died. He claimed to have solutions for a particular function, but only had time to write down a few before moving on to other areas of mathematics. He wrote the following incomplete equation with 14 others, only 3 of them solved.
Within months, he passed away, probably from hepatic amoebiasis. His final notebook was sent by the University of Madras to G. H. Hardy, who in turn gave it to mathematician G. N. Watson. When Watson died in 1965, the college chancellor found the notebook in his office while looking through papers scheduled to be incinerated. George Andrews rediscovered the notebook in 1976, and it was finally published in 1987. Bruce Berndt and Andrews wrote about Ramanujan’s Lost Notebook in a series of books (Part 1, Part 2, and Part 3). Berndt said, “The discovery of this ‘Lost Notebook’ caused roughly as much stir in the mathematical world as the discovery of Beethoven’s tenth symphony would cause in the musical world.”
March 28, 2013 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
RootApproximant can turn an approximate solution into a perfect solution, such as for a square divided into fifty 45°-60°-75° triangles.
A square can be divided into triangles, for example by connecting opposite corners. It’s possible to divide a square into seven similar but differently sized triangles or ten acute isosceles triangles. Classic puzzles involve cutting a square into eight acute triangles, or twenty 1 – 2 – √5 triangles. The last image uses 45°-60°-75° triangles, but one triangle has a flaw.
It’s easy to divide a square with similar right triangles. Can a square be divided into similar non-right triangles? In his paper “Tilings of Polygons with Similar Triangles” (Combinatorica, 10(3), 1990 pp. 281–306), Laczkovich proved exactly three triangles provided solutions, with angles 22.5°-45°-122.5°, 15°-45°-120°, and 45°-60°-75°. I read his paper to try to make an image for the 45°-60°-75° case, but his construction was complex, and seemed to require thousands of triangles, so I tried to find my own solutions. All my attempts had flaws, such as the last image above, so I made a contest out of it: $200, minus a dollar for every triangle in the solution.