Sangaku, Harold Jacobs, and Geometer's Sketchpad

As I mentioned at the time, I delivered a talk at the New England Section of the Mathematical Association of America meeting in Bridgewater, Massachusetts, in November. I decided that it would be good to make the talk available online. The talk was about my adventures in trying to prove a difficult theorem mentioned in Harold Jacobs' Geometry. After finding a proof with the aid of Geometer's Sketchpad I happened to discover through Wikipedia that the theorem has a name: The Japanese Theorem For Quadrilaterals. Then Peter Renz, one of Jacobs' editors, suggested I look at the book Sacred Geometry: Japanese Temple Geometry by Fukagawa Hidetoshi and Tony Rothman, which allowed me to place the theorem in a rich cultural and mathematical context.

The talk is at http://www.scribd.com/doc/129968316/NES-MAA-Presentation. This consists of the slides that I used, put in portrait page orientation, but otherwise unchanged. It is a bit terse, but I hope some find it interesting.

The Power of Cryptograms

When I was a boy, a relative bought me and my siblings a copy of a book about solving cryptograms, which described techniques for solving substitution cyphers and gave a lot of cryptograms of varying difficulty to play with. I really enjoyed the book. At the time, it seems like many newspapers would publish a daily cryptogram, though I don't see them very much any more. My local newspaper, the Boston Globe, publishes a daily crossword, a Sudoku, a Kenken, and a few other puzzles, but no cryptograms. I think that's a loss. The kind of thinking used to solve cryptograms is very similar to what is used in solving mathematical problems of all kinds, and cryptograms is something that can be enjoyed both by mathematically-oriented people and literature-oriented ones, since the quotations that are encrypted can be quite memorable. Also, an interest in simple cryptograms could lead to a long-term interest in cryptography, the importance of which in today's Internet-driven world can scarcely be overestimated.

I'd suggest anyone teaching math in the middle grades think about challenging their students with cryptograms. As a starting point, I found Simon Singh's web page at http://simonsingh.net/cryptography/cryptograms/ to be a nice introduction.

Constructible Angles

A student asked what angles are constructible, considering only angles that measure a whole number of degrees (which I'll call integral angles). The answer is very simple: The only constructible integral angles are those which measure 3n degrees, where n is any natural number.

To see this, first note that angles of measure 60 and 72 (vertex angles in a regular pentagon) are constructible. Since angles may be bisected repeatedly by construction, this means that angles of measure 15 = 60/4 and 9 = 72/8 degrees may be constructed. Since 3 = 9*2 – 15, an angle of 3 degrees may be constructed by constructing two adjacent 9 degree angles, and then a 15 degree angle inside the resulting 18 degree angle. By adding n 3-degree angles together, a 3n degree angle can be constructed.

The proof that only these integral angles can be constructed seems to require the more advanced result that there is at least one integral angle that is not constructible. For example, an angle of 20 degrees is non-constructible. (This is the standard proof, using Galois theory, that demonstrates the impossibility of angle trisection.) If there were an integral angle of measure n that is not a multiple of 3 that is constructible, then since (n,3) = 1, there is a linear combination with integral coefficients of n and 3 that gives 1, and so an angle of 1 degree would be constructible, and hence an angle of 20 degrees would be constructible, which it is not..

Wallace Feurzeig 1927 - 2013

Computer pioneer, mathematician, educator, mentor, and friend Wally Feurzeig passed away on January 4. Wally was a key member of the small  team that created the LOGO educational programming languages in the 60s. I' was privileged to know him and his lovely wife Nanni for over 15 years. Wally would listen patiently anc carefully to my various enthusiasms about math education, and always find something incredibly helpful to say, or introduce me to others from his circle such as Victor Gutenmacher and the late Oliver Selfridge who really expanded my understanding of what mathematics education could be. My wife Leslie and I were grateful for invitations to a Monday evening dinner that was cooked by his lovely wife Nanni, and featured some of their fascinating friends.

You can read about some of Wally's accomplishments on Wikipedia, but I just wanted to record a few personal remembrances here.

I am feeling the loss of a kind, gentle, and most intelligent man. We will not see Wally's like again.

The Four Kinds of Students

Richard Feynman is generally regarded as a great teacher, and I'd agree, based on my experience when I had him in sophomore physics at Caltech. However, he sometimes despaired of the teaching enterprise. He said something to the effect that teaching a concept is either unsuccessful (in the case of a poor student) or unnecessary (in the case of a good student who can pick it up by reading). I'd like to suggest that there are not just two, but four kinds of students.

The poor student is unable or unwilling to learn the material.

The mediocre student will learn as much as the teacher presents, but no more.

The good student will continue to learn after leaving the teacher.

The excellent student will surpass his or her teacher.

A Sangaku Problem

I will be giving a talk Saturday at the NES-MAA meeting in Bridgewater MA in which I will talk about sangaku, or Japanese temple geometry problems. These problems, created by people from a wide walk of life, were beautifully drawn on wooden tablets which were then placed in Buddhist temples or Shinto shrines. Hundreds have been discovered, and probably thousands existed at one time. I became involved in this when I was challenged to solve one of the difficult sangaku. Here I present one of the easier ones, from the delightful book Sacred Geometry: Japanese Temple Geometry by Fukagawa Hidetoshi and Tony Rothman.

I chose this problem because the diagram is so beautiful, the solution is fairly simple, yet satisfying, and it is one of the few sangaku created by a woman (Okuda Tsume).

In a circle of diameter AB = 2R, draw two arcs of radius R with centers A and B respectively, and 10 inscribed circles, two green circles of diameter R, four red circles of radius t, and four blue circles of radius t'. Show that t = t' = R/6.

In the diagram below, we follow the convention of labeling the center of a circle with the radius of that circle.

E22. A simple probability problem

How many rolls of one fair die does it take, on average, before all six numbers show up? Make a guess and then see if you can figure it out.