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- Creating Symmetry: The Artful Mathematics of Wallpaper Patterns by Frank A. Farris
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Keywords: book review ; color symmetries. Crystallographers are probably most familiar with tilings, uniformly discrete sets of points, or algebraic systems of vectors and tensors. On occasion, one encounters periodic energy landscapes, but that is not how crystallography is introduced. Frank Farris introduces two-dimensional crystallography using functions from the plane to the plane — more precisely, from the complex plane to the complex plane.
There is a target photograph, as in Fig. The wavefunction maps the plane to the photograph, and the plane left of the photograph is colored so that every point's color is the color of its image point on the photograph. The result is that the domain of the wavefunction has a pattern induced by the wavefunction, and if that function has been set up properly, that pattern is crystallographic.
This example appeared in the discussion of color symmetries. Notice that the purple and green regions are bounded by black curves, which means that all those blackened points are mapped to the black line down the center of the photograph. In all the periodic examples, the entire plane is mapped repeatedly into a portion of the photograph. Here is how the patterns are constructed. Associating a point with a complex number , a function from 2-space to itself — viz. Suppose you colored the plane as in Fig. The origin is in the white center, the real axis is horizontal so that 1 is colored red while the imaginary axis is vertical so that i is on the boundary between green and yellow.
If we colored the plane to show where this wavefunction f sends points, we would get the image on the left of Fig. The primary construction in this book is to convert symmetries into formulas for complex functions.
A symmetry of a function is a function such that for every ,. For example, the complex conjugacy function often denoted captures reflection across the real axis, so that for all if and only if the pattern of f has the horizontal axis as a mirror. For example, Farris chose which he did not reveal to obtain a function. Complex analysis usually considers those functions for which the derivative as a limit.
Such functions are called analytic , and the peculiar properties of analytic functions are what makes complex analysis like a magic act.
But most of the periodic functions in the book are not analytic, and thus much of the machinery of complex analysis cannot be applied to them. Then look at curves A and B below and see if you can match them to the correct pairs of numbers, from the 10 choices given.
Creating Symmetry: The Artful Mathematics of Wallpaper Patterns by Frank A. Farris
Give your reasons for choosing your particular pairs and describe what you think the relationship is between the two numbers. How are the numbers related to the curves, and how can we generate other similarly graceful curves?
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Our next two challenges arise from these questions. Question 2 requires some mathematical sophistication, but you can skip it if you wish and go straight to question 3, which only requires a willingness to explore and play, and create beauty. The curve has a fivefold symmetry, which you can clearly make out in the image at right. The mystery is this: What do the coefficients 6 and 14 have to do with generating fivefold symmetry? Why did we use —14 instead of 14 in question 1? Can you explain how this formula works to a general audience? Anyone can generate the above curve — use your favorite graphing app, or enter the above expression into Wolfram Alpha or the excellent graphic calculator Desmos.
Now start playing with the parameters — change the coefficients, fiddle with the signs, do anything you want using sine and cosine terms. But who knows what a mathematician does?! While it is certainly a textbook, Creating Symmetry Princeton University Press, is a beautiful book, worthy of coffee table status. It can be read on many levels and is accessible to a wide range of readers. It also does an amazing job of offering a glimpse at the work of a mathematician.
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Each chapter focuses on a question along the theme of incorporating interesting symmetry in design. Beginning with a basic analysis of curves chapter 1: How do you make a circle?
By the end, he presents everything from wallpaper groups to hyperbolic geometry, keeping an eye on aesthetics all along the way. Creating Symmetry would be great for a seminarstyle class or a challenging undergraduate course.
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It tells an interesting story about aesthetics and beauty in mathematics while tying together much of the math in the undergraduate curriculum: calculus, geometry, abstract algebra, complex analysis, and more. Also, it contains interesting and challenging exercises, and enough mathematics to keep students busy for months.