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Forbidden Symmetries Frank A Santa Clara University Scholar Commons Mathematics and Computer Science College of Arts & Sciences 11-2012 Forbidden Symmetries Frank A. Farris Santa Clara University, [email protected] Follow this and additional works at: http://scholarcommons.scu.edu/math_compsci Part of the Mathematics Commons Recommended Citation FARRIS, Frank A. "Forbidden Symmetry,"Notices of the American Mathematical Society, November 2012: 2-6. (Lead article and additional cover image.) First published in Notices of the American Mathematical Society in v.59 no.10, 2012, published by the American Mathematical Society. This Article is brought to you for free and open access by the College of Arts & Sciences at Scholar Commons. It has been accepted for inclusion in Mathematics and Computer Science by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Forbidden Symmetries Frank A. Farris he crystallographic restriction, as it applies to patterns in the plane, tells us that a pattern invariant under two linearly independent isometries can- not have 5-fold symmetry. And yet the Tpattern in Figure 1 seems to have translational symmetry in two directions as well as rotational symmetry through 72◦. To see what I mean, start with the wheel shape at the point labeled A and notice translational symmetries along vectors AB→ and AC→; then rotate the image 72◦ about A, so that B goes to C, and see that the image is unchanged, apparently in violation of the crystallographic restriction. How can this be? A moment’s thought can break the illusion: If the pattern really enjoyed translational symmetry along vector AB→, then we could rotate 72◦ clock- wise about B and move A to C. Alas, angle ABC ∠ is an unfortunate 54◦, so B is not truly a translate Figure 1. Wallpaper with 5-fold symmetry? of A. The purpose of this paper is to show that an effort to construct functions known not to displayed 5-fold symmetry and so fell outside the exist may on occasion produce interesting frauds. mathematical categories commonly accepted as Our method produces a family of Harald Bohr’s encompassing all possible crystalline structures. quasiperiodic functions, which may well remind Our quasiperiodic functions are a different sort readers of the quasicrystals that have been much of object altogether. Unlike the structures studied in the news since Daniel Shechtman won the Nobel by crystallographers, which are idealized as sets Prize in Chemistry in 2011 [1]. of isolated points in space, these are smooth func- The term quasicrystal has an interesting his- tions that have honest 5-fold symmetry about a tory, as explained by Senechal in the Notices [6]. single point and come so close to having transla- Diffraction patterns found by Shechtman in 1982 tional symmetry that they can easily fool the eye, Frank A. Farris is associate professor of mathematics at provided we select a suitable translation. Santa Clara University. His email address is ffarris@ The techniques developed to create these func- scu.edu. tions may offer more interest beyond creating DOI: http://dx.doi.org/10.1090/noti904 fraudulent images such as Figure 1. We show how 1386 Notices of the AMS Volume 59, Number 10 to construct functions with 3-fold symmetry and For 3-fold symmetry, we may either take such how the technique breaks down when we try to an f and average it over cyclic permutation of change 3 to 5. variables or require that an anP anP 2 , Preliminaries = = since these coefficients will become equal after By wallpaper group we mean a group G of averaging. (Here, we think of the matrix P as Euclidean isometries of the real plane whose acting on row vectors n, the adjoint action.) translational subgroup is a lattice generated by It remains to find planar functions somewhere two linearly independent translations. As is well in this 3-dimensional setup. The eigenspaces of known, there are seventeen isomorphism classes the linear transformation P are easily found to be of such groups. Also well known is the fact that the line generated by [1, 1, 1] and a plane , with if ρ is a rotation in one of these groups, then its basis V1 [1, 1, 0] and V2 [0, 1, 1], onΠ which = − = − order is 2, 3, 4, or 6. P acts as rotation through an angle 2π/3. Unlike many discussions of patterns in the If we restrict f to the plane , the resulting plane which refer to subsets, called motifs, being function has translational invarianceΠ with respect repeated without overlap [4, p. 204], we use anal- to the integer vectors V1 and V2, as well as ysis to develop the concept of pattern. For us, a rotational symmetry through an angle of 2π/3. pattern is given by a wallpaper function, which is Of course, f is necessarily invariant with respect a real- (or perhaps complex-) valued function on to all compositions of these symmetries and so the real plane that is invariant under the action is a wallpaper function with group at least as of one of the wallpaper groups [2]. In symbols, we large as the one generated by the rotation and one require that translation. (Translation along V2 can be written by conjugating the other translation by the rotation.) 2 f (gx) f (x) for every x R , g G. We call this group p3, preferring the notation = ∈ ∈ of the International Union of Crystallography to For any given group G, it is easy to con- Conway’s orbifold notation. struct such functions by superimposing plane We can easily construct wallpaper functions waves invariant with respect to the lattice of with groups that contain p3 as a subgroup by translations in the group. My paper “Wallpaper including special groupings of terms in the Fourier functions” [2] explains the construction and also expansion of the function of three variables before covers functions with color-reversing symmetries. we restrict. To explain this in some detail, we begin For wallpaper functions with 3-fold symmetry, an with packets of waves invariant under 3-fold alternate method is possible, one that readers fa- rotation: miliar with group representations may recognize. 1 2πin x 2πin Px 2πin P 2x Wn(x) (e · e · e · ), We present this method and try to generalize it = 3 + + to produce functions with 5-fold symmetry. In the where we divide by three simply so that Wn(0) 1. = generalization, we can see exactly why Figure 1 Notice that when restricted to the plane , fails to enjoy honest translational symmetry. Π Wn (j,j,j) Wn for j N. + = ∈ Constructing Patterns with 3-fold Symmetry Setting n (n1, n2, n3) and representing a typ- = ical point in as XV1 Y V2, we write these wave To construct functions on the plane with 3-fold + symmetry, we start with an unlikely object: cyclic packets in theΠ form permutation of three variables, considered as 1 2πi(nX mY ) 2πi(mX (n m)Y ) Wn,m(X, Y ) (e + e − + a linear transformation of R3. After all, this = 3 + 2πi( (n m)X nY ) permutation, which we will call P, does have e − + + ), + order 3. If f is any function of three variables, where then f (x, y, z) f (z, x, y) f (y, z, x) is a function + + n n1 n2 and m n2 n3. invariant under P. = − = − If we wish to choose n and m first and find a Since we seek periodic functions, let us suppose vector n that gives rise to those frequencies, one that the function f (x, y, z) is periodic with respect choice is (n m, m, 0). to the integer lattice in R3. Let us limit ourselves + This is why we claim that every (sufficiently to constructing continuously differentiable func- smooth) wallpaper function on the plane with 3- tions. Then we may assume that f is represented fold rotational symmetry can be exhibited as the as the sum of its Fourier series in three variables: restriction of a function of three variables that is 2πin x periodic with respect to the integer lattice and in- f (x, y, z) ane · . = n Z3 variant under cyclic permutation of variables: The ∈ November 2012 Notices of the AMS 1387 functions Wn,m(X, Y ) form a basis for functions consisting of vectors with integer coordinates. with the desired symmetry in the plane, and every ThisΠ is the only reason we could claim that the Wn,m is the restriction of some Wn. We write restricted function is periodic with respect to a lattice within , given only that it is periodic with (1) f (X, Y ) a W (X, Y ) n,m n,m Π R3 = n,m respect to the integer lattice in . for the typical wallpaper function—a superposi- What Happens If We Use 5 Variables? tion of wallpaper waves. R5 To continue the discussion of functions with The same construction can be attempted in . Let additional symmetries, we omit some messy de- us review why we do not expect to create functions tails and present the reader with the fact that the with 5-fold symmetry that are also periodic with map respect to a lattice. The reason to include this well- known explanation is that it suggests something σc (X, Y ) (Y , X) = to look for in the images. is a mirror reflection in , so that any sum of the form (1) where Π The Crystallographic Restriction an,m am,n = will represent a function with mirror symmetry. We prove a special case of the crystallographic Every function of the form restriction, showing that 5-fold rotations are never present in wallpaper groups: Suppose a wallpaper f (X, Y ) an,m(Wn,m(X, Y ) Wm,n(X, Y )) = + group G contains a translation T and a rotation R n,m through 2π/5 radians.
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