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Physical Models for Non-Euclidean Geometry

I strongly believe in the use of physical models, whenever possible, to introduce mathematical concepts. For example, when teaching non-Euclidean geometry to high school teachers, I like to have them create triangles on actual physical spheres, using rubber balls, push pins, and rubber bands to create geodesics. It is easy to “discover” that the sum of the angles in spherical triangles is greater than 180 degrees, and that the excess of a triangle (the sum of the angles minus 180 degrees) is additive and hence proportional to the area of the triangle. These demonstrations can be easily done for specialized triangles, so the student becomes familiar with the geometric fact before thinking about how it might be proven.

What I would like to do next is to show that the sum of the angles in a triangle on a surface of negative curvature is less than 180 degrees. This leaves me with two problems.

(1) How can I make (or obtain) a physical model of a simple saddle surface for experimentation by students. Ideally, models should be cheap enough so that I can supply each pair of students with a model to work with.

(2) How can students draw geodesics on such a surface? Rubber bands are not going to work here, because a band stretched between two points on the surface will not necessarily lie on the surface. This problem is sort of mathematical, because I think a good understanding of the nature of geodesics should lead to discovering a way of having students create them on a surface of negative curvature.

I am aware of some very good software that uses the Poincare disk model to do geometry on the hyperbolic plane, and I plan to use the software when I teach. But I want students to have real physical experience first.

Anyone have any ideas?

Visual Calculus

I found a fascinating page: VisualCalc. This is a talk by Tom Apostol about Visual Calculus, a technique for finding the area bounded by curves without using traditional calculus developed by an Armenian mathematician living in California, Mamikon A. Mnatsakanian, which has been espoused by Apostol. Some of the results using this method would be very difficult if not impossible to uncover with traditional methods. The starting point is the following simple (but neat) problem, solved by Mamikon (as he calls himself) when he was 15:

Problem: A line segment is drawn tangent to the inner of two concentric circles, terminating at the outer circle. The length of the segment is 2a. What is the area of the annulus?

Answer: pi*a^2. It is rather counterintuitive that the result is independent of the radius of the inner circle.

Solution: Let the radius of the smaller and larger circles by r and R, respectively. The area of the annulus is pi*(R^2 - r^2). Draw the obvious right triangle with legs of length r and a, and hypotenuse of length R. Apply the Pythagorean Theorem.

Mamikon noted that if he knew in advance that the answer was independent of r, he could let r = 0, and the tangent segment would become a diameter of the larger circle, establishing the result another way. This led him to a rather breathtaking extension of the result.

Theorem 1. Let C be a smooth convex oval. Move a vector v (of fixed length) around the oval (with the tail on the curve) so that it is always tangent to the curve (at its tail). Then the area swept out by the vector is pi*|v|^2. [I'm not sure what the exact hypothesis is, but this is the basic idea.]

Proof idea: Let S be the set of translates of the vectors v(t), with a common tail formed as v(t) goes around the oval. Then S is a circle of radius |v|.

Tri-Color chessboards

When coloring a checkerboard, the basic requirement is that squares that are full-neighbors (horizontally or vertically) have different colors. Clearly, there are exactly two ways of coloring an n x n checkerboard with two colors (black and red, say). Once a color has been selected for the lower left corner, all remaining square colors are forced. I wondered how many different ways one could color an n x n checkerboard with three colors. This led me to consider two problems:

(1) How many ways are there to color an n x n checkerboard, using at most 3 colors?

(2) How many ways are there to color a 3k x 3k checkerboard, using equal numbers of red, blue, and white squares?

Bridget Tenner of dePaul University immediately came up with the answer to problem (1) by searching in Neal Sloane's wonderful Online Encyclopedia of Integer Sequences, using "3-color" as a search string. The answer is given here as a special case of A078099 (for m x n checkerboards), which is defined recursively. The sequence grows very quickly: it is 3 times a sequence beginning 1, 6, 82, 2604, 193662, 33865632, 13956665236.

A sequence to answer question (2) does not seem to appear in OEIS, so this may be an open question.

Three Books on Riemann Hypothesis

My first review will be of three semi-popular books about the Riemann hypothesis: Prime Obsession by John Derbyshire, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus Du Sautoy, and The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics by Karl Sabbagh.

Recently there has been a spate of semi-popular books about the Riemann Hypothesis. This is doubtless due in part to the fact that several of the most famous problems of modern mathematics such as the Four-Color Map Theorem, Fermat's Last Theorem, and the Poincare Conjecture have now been solved, leaving the Riemann Hypothesis as the most famous problem standing. However, writing a semi-popular book about the Riemann Hypothesis is an intimidating mission. Unlike the Four-Color Map Theorem and Fermat's Last Theorem, it is difficult to explain to an educated layperson what the theorem states, or why it is important. Even the statement of Poincare Conjecture is easier to comprehend.

John Derbyshire's Book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, is the best of the lot. He sets himself the daunting task of explaining virtually all of the major mathematical ideas needed to understand the statement of the RH, its relation to the distribution of prime numbers, and some of the major methods that have been used to attack the problem, in a book designed for an otherwise educated person who is ignorant of mathematics from high school algebra on. It sounds to me that this goal must have been imposed the publishers, because whatever the talents of the expositor, it is prima facie impossible to bring anyone but a latent mathematical genius on such a trip in the confines of a single 422-page book. However, what Derbyshire does, and does brilliantly, is to explain the RH to someone who has understood two years of college calculus, or the equivalent. The reader who has experience with integrals and infinite series should be able to follow the exposition.

The Reimann Hypothesis and its relation to the distribution of primes belongs to the branch of mathematics called analytic number theory. This subject is not easy to write about. I took a reading course in analytic number theory in graduate school. I was intrigued by the subject, but became discouraged when I found the text, by a famous researcher in the field (who shall remain nameless) riddled with errors. I ended up going into another specialty. Now that I have read Derbyshire's book, I'm tempted to read more. In addition to the mathematical exposition, Derbyshire quickly and deftly sketches the political and social milieu and the personalities involved in the development of the RH and the search for its solution.

I bring up one quibble because it relates to the first chapter, and might cause a reader to give up. Derbyshire introduces the harmonic series (and its divergence) by asking the reader to imagine constructing a bridge out of playing cards. It turns out that the n-th card from the top of this bridge can extend 1/(n – 1) card length from the card above it, so that the span of the entire n-card bridge is 1 + 1/2 + … + 1/n. I've seen this before, and it is cute, but it is not easy. There are easier ways to introduce the harmonic series.

Marcus du Sautoy's book, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, covers much of the same territory as Derbyshire's book, but goes into somewhat less mathematical detail. Du Sautoy is a professor of mathematics at Oxford, and an excellent writer. I recommend this book for the poetry of the language and the vividness of the stories of the mathematicians involved in the story. It is wonderful to read a book by a first-rate mathematician who is also a first-rate storyteller.

In Karl Sabbagh's book, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, the author, like John Derbyshire, attempts to explain the RH to the mathematically unsophisticated reader. In this case, the mathematical basics are covered in a series of appendices, called "Toolkits". This book was something of a disappointment. First, the author comes across as a journalist rather than as an advanced amateur mathematician (like Derbyshire) or a professional mathematician (like du Sautoy). The writing has more of a superficial feel to it, where more tends to be made of the physical appearance or personal idiosyncrasies of mathematicians rather than their ideas. In addition, Sabbagh spends much of the book conversing with and about Louis de Branges, who has claimed to have a proof of the Riemann Hypothesis. It is true that de Branges is a respected mathematician who solved an important long-standing problem, the Bieberbach Conjecture. However, very few mathematicians credit his claims to have, or be close to, a proof of the Riemann Hypothesis. Sabbagh is obviously charmed by de Branges, and spends, in my opinion, far too much time on this player who seems to deserve at most a short footnote in the story.