Finite Plane Symmetry Groups Joseph A. Gallian University of Minnesota, Duluth, Duluth, MN 55812 In refs I and 2 Hungarian needlework was used to illustrate the frieze groups and the plane crystallographic groups. Others have used African, Chinese, Indian, and Islamic art to illustrate the same families. But what about finite groups of symmetries? What are rich sources for illustrating cyclic and dihedral groups through symmetry? In
Figure 2. Hubcaps with symmetry groups L k h,Ds, Dm, 4 8 ,
their delightful book on symmetry, Hargittai and Hargittai (3) use flowers, postage stamps, corporate logos, and even coins to illustrate these groups. In contrast, my favorite source is hubcaps! Hubcaps are preferable to exotic samples of artworks, crafts, flowers, stamps, and coins because students encounter diverse assortments of hubcaps daily. A
!&. Volume 67 Number 7 July 1990
549
stroll through a parking lot becomes an adventure in group theory. Automobile hubcaps and wheels offer a broad array of finite symmetry groups, provided one makes allowances for imnerfections such as the notch for the valve stem. the automobile name or logo, and the area where the wheel lugs are. RecallLeonardo davinci's theorem (4): the only finite plane symmetry groups are C, and D,, the cyclic group of order n and the dihedral eroun of order 2n. Ohviouslv hubcavs are three-dimension&, hui the orthogonal projection of hubcavonto a plane (ex., the vhotopra~hicimage)is isomomhic to-the three-dirnezonai symmetry of the hubcap itself. (A reflection across a plane in three dimensions corresponds to a reflection across a line in the orthogonal projection.) In one brief foray to a few parking lots I netted hubcaps (or wheels) with C, symmetry groups for n = 2,5,9,10,
a
550
Journal of Chemical Education
12,15,17,18,20,22,24, andD. symmetry groups for n = 2,3, 4,5,6,7,8,10, 12, 13,15,16, 18,20,24,30,32,34,36, and 72. Subsequent searching turned up hubcaps with symmetry groups C, for n = 3,4,6,7,8,14,16,31, and 60 and D, for n = 28,35,40,76,80, and 100. Some of these are shown in Figures 1and 2. Literature Cited
1986. I . Gsllisn, J. Contempomry Abstract Aimbm: Heath: Lexington, MA, 1986; p 330
This is an abridged version of an article, "Symmetry in Logos and Hubcaps," appearing in the Amerlcan Mathematical Monthly 1990 97, 235.
