Bruce Torrence

The number of edges in a subgraph of a Hamming graph

Abstract Let G be a subgraph of the Cartesian product Hamming graph (Kp)r with n vertices. Then the number of edges of G is at most ( 1 2 )(p − 1) logp n, with equality holding if and only G is isomorphic to (Kp)s for some s ≤ r.

The Fundamental Theorem of Algebra for Artists

I t’s simple. It’s beautiful. And it doesn’t get more fundamental: Every polynomial factors completely over the complex numbers. So says the fundamental theorem of algebra. If you are like us, you’ve been factoring polynomials more or less since puberty. What may be less clear is why. The short answer is that polynomials are the most basic and ubiquitous functions in existence. They are used to model all manner of natural phenomena, so solving polynomial equations is a fundamental skill for understanding our world. And thanks to the fundamental theorem, solving polynomial equations comes down to factoring. However, finding the factors is often easier said than done. For instance, start with your favorite polynomial equation, something like Any such equation can be rewritten so that it is equal to zero; in this case we add one to each side: The fundamental theorem says the polynomial on the left factors completely. With a bit of work we get Since a product of numbers is zero if and only if one of those numbers is itself zero, the factored form tells us the solutions to the original equation: and But wait. What about It can’t be factored, you may protest, since a square plus one cannot be zero. While this is true for real numbers x, this polynomial (and according to the fundamental theorem, all polynomials) does factor over the complex numbers. In other words, to factor mathematicians first had to ask: Is there a larger domain of numbers where it could be factored? And after a long time—several centuries into the enterprise of solving polynomial equations—the complex numbers were discovered. These are numbers of the form where a and b are real numbers and is an “imaginary” number satisfying Complex numbers can be thought of as points in the Euclidean plane by associating the number with the ordered pair So complex numbers turn points in the plane into numbers to which we can apply the four elementary operations of addition, subtraction, multiplication, and division. This was a profound discovery. In this case, we have We will be leaving the comforts of the real number system behind, so let’s agree to use z rather than x for the variable name. It’s a time-honored notational convention that z represents a complex variable. Whether over the real or complex numbers, factoring is hard. Given n complex numbers (think of them as points in the complex plane), it is a cinch to construct a polynomial with those roots: Just expand the product But suppose we are given a polynomial of degree n, that is, a function of the form

Fibonacci, Lucas, and a Game of Chance

Summary A simple game of chance is introduced, where it is shown that each players winning probability is a ratio of Fibonacci and Lucas numbers.

MATHEMATICS MEETS PHOTOGRAPHY PART II: Conformal Is the New Normal

In part I of this series, we described photographic panoramas that capture the camera’s entire surroundings. Not just front, back, left, and right, but also up and down. These panoramas capture the viewable sphere, an imaginary ball around the camera with imagery on it that matches the surrounding scene. To display spherical panoramas on a flat surface, we borrow ideas from cartography, the science of mapping a sphere to a plane. A certain class of these mappings, or projections, is well suited to displaying visible spheres: conformal mappings—functions that preserve angles when mapping imagery from one domain to another. Conformal mappings work well because although there may be some large-scale distortions, small details do not get skewed or stretched into an unrecognizable mess. We finished part I by describing a particular conformal mapping: the stereographic projection, which entails treating the viewable sphere as a translucent ball with a light bulb on top, from which the imagery is projected onto the floor. The result is a striking image that looks like a little planet floating in the sky. See figure 1.

The Student's Introduction to Mathematica ®: Working with Mathematica

Practical information to and tips for using Mathematica and the Wolfram Language. Document creation, slideshow presentations, keyboard shortcuts, documentation, and troubleshooting are discussed.