Music from Vibrating Wallpaper

Wallpaper patterns have been shown to be decomposable into standing waves of plane vibrations [6]. Previously unexplored are the sounds that arise from these vibrations. The main result of this paper is that each wallpaper type (square, hexagonal, rectangular, generic) has its own distinctive family of pitches relative to a fundamental. We review the method to make wallpaper with wave functions and describe new musical scales for each type, including initial attempts to use the scales: a movie showing vibrations of wallpaper patterns with 3- and 6-fold symmetry inspired a new piece by American composer William Susman, commissioned by the San Jose Chamber Orchestra, Barbara Day Turner, conductor. The piece, “In a State of Patterns,” was premiered on March 25, 2018

The Editor\u27s Song

The program of the 2011 Mathfest\u27s opening banquet was MAA-The Musical!\u27 Produced by Annalisa Crannell and starring the MAA Players (active MAA members all), it highlighted activities of the Association and of Mathfest itself. This song represents the journals. It was sung by past editor Frank Farris to the tune of A Wand\u27ring Minstrel I, from Gilbert and Sullivan\u27s the Mikado

An intrinsic construction of Fefferman\u27s CR metric

We construct a conformal class of Lorentz metrics naturally associated with an abstract definite CR structure. If the CR structure is that of a pseudoconvex boundary in Cn we prove that the intrinsically constructed metric is the same as that discovered by Fefferman using a solution to a complex Monge-Ampère equation. The construction presented here relies on formal solutions of a linear equation, dζ = 0, and provides a relatively simple procedure for computing the metric

Wheels on Wheels on Wheels-Surprising Symmetry

While designing a computer laboratory exercise for my calculus students, I happened to sketch the curve defined by this vector equation: (x, y) = (cos(t), sin(t)) + 1/2(cos(7t), sin(7t)) + 1/3(sin(17t), cos(17t)). I was thinking of the curve traced by a particle on a wheel mounted on a wheel mounted on a wheel, each turning at a different rate. The first term represents the largest wheel, of radius 1, turning counter-clockwise at one radian per second. The second term represents a smaller wheel centered at the edge of the first, turning 7 times as fast. The third term is for the smallest wheel centered at the edge of the second, turning 17 times as fast as the first, clockwise and out of phase

The Edge of the Universe-Noneuclidean Wallpaper

A simple mathematical model of a two-dimensional universe, called the Poincaré Upper Halfplane, illustrates the possibility of a universe with an unattainable edge. In this article, I describe this model -a famous example of a noneuclidean geometry-and explain how conversations with an analytic number theorist led me to create wallpaper patterns for its inhabitants. These are interesting not only for their high Gee whiz! factor, but also as a window for observing the features of this unusual geometry

Review of Visual Complex Analysis, by Tristan Needham

Tristan Needham\u27s Visual Complex Analysis will show you the field of complex analysis in a way you almost certainly have not seen before. Drawing on historical sources and adding his own insights, Needham develops the subject from the ground up, drawing attractive pictures at every step of the way. If you have time for a year course, full of fascinating detours, this is the perfect text; by picking and choosing, you could use it for a variety of shorter courses. I am tempted to hide the book from my own students, in order to appear the more clever for popping up with crisp historical anecdotes, great exercises, and pictures that explain things like that mysterious 2πi that crops up in integrals. Whether you use Visual Complex Analysis as a text, a resource, or entertaining summer reading, I highly recommend it for your bookshelf

Crafty Counting

To count a set means to put it in one-to-one correspondence with a set of integers {1, 2, 3,...,n}. Direct counting is nice, but in complicated situations it pays to be more crafty. A problem with patterns of colored tiles gives us a chance to illustrate a popular counting principle known by various names. We\u27ll call it the Burnside-Cauchy-Frobenius formula. It is also popularly called the Burnside Orbit-Counting Lemma, though wags refer to it as not Burnside, because it was known long before Burnside was born. Later Pólya generalized the formula, so some readers may recognize this as Pólya Enumeration

A Window on the Fifth Dimension

Is there enough mathematics in your home? What visual aids do you keep on hand for that inevitable moment when guests want to know why you spend your life on mathematics? Feeling a lack in this area, I commissioned glass artist Hans Schepker to produce a window - from the fifth dimension? - based on an image that came up in my research. It turned out splendidly, and you can see it on the cover of this issue of MAA FOCUS

Forbidden Symmetries

The purpose of this paper is to show that an effort to construct functions known not to exist may on occasion produce interesting frauds. Our method produces a family of Harald Bohr\u27s quasiperiodic functions, which may well remind readers of the quasicrystals that have been much in the news since Daniel Shechtman won the Nobel Prize in Chemistry in 2011

Writing Mathematics-A Nut and a Bolt of Style

As editor of Mathematics Magazine, I see a lot of manuscripts. Some of them are written with a charming sense of style, but many of them leave me thinking that the author\u27s only concern was to set out the mathematics clearly. This is a fine place to start, but the tradition of the Magazine is to offer things that people will enjoy reading, and this requires more than clarity. Let me explain an important step authors can take in order to make their work more attractive