The Rotating Disk

Figure 106 A

Figure 106 A

D

D. St. P. Barnard's game problem

3. FRIEZE PATTERNS

A frieze is a pattern that endlessly repeats itself along an infinite strip. Such patterns can exhibit different kinds of basic symmetry, but here we shall be concerned only with what is called "glide symmetry." A glide consists of a slide (more technically a "translation") combined with mirror reflection and a half-turn. For example, repeatedly gliding the letter R to the right along a strip generates the following frieze:

RBRBRBRBRBRBRBR...

H. S. M. Coxeter, a geometer at the University of Toronto, recently investigated in depth a remarkable class of frieze patterns that can be constructed very simply by using nonnegative integers, if the lack of symmetry in the shapes of the numerals is ignored [see Figure 107], Think of the numerals as representing spots of colors, all l's the same color, all 2's another color and so on. In this instance any rectangular portion of the frieze that is nine columns wide, such as the shaded one shown here, can be regarded as the unit pattern. By gliding it left or right— that is, sliding and simultaneously reflecting and inverting— the infinite frieze pattern is generated.

Figure 107

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1 1 vMliPHBBilHHtBift1 1

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A frieze pattern with glide symmetry

To produce this type of frieze pattern, begin with infinite borders of O's and l's at top and bottom, and a path of numbers from top to bottom such as the zig-zag path of eight l's shown on the left between the borders of O's. The numbers in such a path (which may be straight, or crooked as it is here), as well as the length of the path, can be varied to produce different patterns. A simple formation rule, common to all such patterns, is now applied to obtain all the other integers. The surprising glide symmetry that results is a nontrivial consequence of this rule.

Our puzzle, suggested by Coxeter, is to guess the simple rule. Hint: It can be written as an equation with three terms involving nothing more than multiplication and addition, and no exponents. When Coxeter first showed the pattern given here to the mathematician Paul Erdos, Erdos guessed the rule in 20 seconds.

A discussion of the properties of such friezes, their fascinating historical background and their applications to determinants, continued fractions and geometry can be found in Cox-eter's "Frieze Patterns" in Acta Arithmetica, Volume 18 (1971), pages 297—310. On friezes in general and their seven basic kinds of symmetry see Coxeter's modern classic, Introduction to Geometry (Wiley, 1961), pages 47-49.

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