Creating Symmetry: the Artful Mathematics of Wallpaper Patterns . By Frank A. Farris. Princeton University Press, 2015. Pp. 248. Price GBP 27.95, USD 35.00 (hardcover). ISBN 9780691161730.

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. The text uses these ‘wavefunctions’ to describe wallpaper symmetries, distorting photographs to get esthetically interesting crystallographic patterns – in a book intended to attract mathematically proficient undergraduate students to crystallography. Here is how it works. There is a target photograph, as in Fig. 1. 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. The complex plane is the set C = fxþ iy: x; y 2 Rg = fr cos þ ir sin : r 0 and 2 1⁄20; 2 Þg, where i 1⁄4 ð 1Þ [so that Euler’s formula gives us expði Þ = cos þ i sin ] and r = ðx þ y2Þ. Associating a point ðx; yÞ 2 R with a complex number xþ iy 2 C, a function from 2-space to itself – viz. from the plane to a photograph – can be represented by a function from C to C. Suppose you colored the plane as in Fig. 2 (right image); this is an example of a ‘color wheel’. 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). The polynomial f ðzÞ 1⁄4 z maps a point z = rðcos þ i sin Þ to a point z = r1⁄2cosð2 Þ þ i sinð2 Þ , doubling its angle while pushing it away from the unit circle. If we colored the plane to show where this wavefunction f sends points, we would get the image on the left of Fig. 2. Heading towards periodic patterns, if gðzÞ = gðxþ iyÞ = expð2 iyÞ = cosð2 yÞ + i sinð2 yÞ, we get Fig. 3 (on the left): a point ðx; yÞ is mapped to expð2 iyÞ = cosð2 yÞ + i sinð2 yÞ, so we get horizontal bars. For example, starting from 1 = 1 + 0i in the middle of a red bar [as it is mapped to 1 = 1 + 0i in Fig. 2 (right)], i is on the next boundary up between yellow and green, as that is where g maps i = 0 + 1i. Farris exhibits a function (which happens to be hðzÞ = ð1=3Þfexpð2 iyÞ + exp1⁄22 ið3x yÞ=2 + exp1⁄22 ið ð3Þx yÞ=2 g) to obtain a periodic coloring of the domain, the color of a point indicating where in Fig. 2 (right image) the point was sent. Now, h(0) = 1 (so that the origin is colored red) while h(1) = h1⁄21=2þ ið3Þ=2 = . . . h1⁄21=2 ið3Þ=2 = 0 (so that 1 is colored white), and repeating for other points we obtain Fig. 3 (right). The primary construction in this book is to convert symmetries into formulas for complex functions. A symmetry of a function f :C! C is a function :C! C such that ISSN 2053-2733