Frieze Tables

Mathematical Connections

Frieze Groups

Elements of the Downtown (the Bakery, Conway’s Curios and the Terrace) illustrate the wallpaper groups, the seventeen different symmetry types for a pattern that repeats in two or more directions in the plane. For a pattern that repeats in only one direction, the seven different symmetry types constitute the frieze groups.

John H. Conway, after whom Conway’s Curios is named, proposed labels for the frieze groups based on varieties of silly walk: hopping, stepping, sidling, jumping, and so forth as pictured here.

Those seven are the one-color frieze groups, in which the basic motif can be simply repeated (HOP) or mirrored (JUMP/SIDLE). rotated (SPIN), and/or glide-reflected (STEP) before being repeated. When mice reproduce the motif with two different colors, each of these moves can be combined with a color change, giving them reflections, rotations, and glide reflections that swap the two colors (black and white), and others that leave the colors alone. Now, the mice have their choice of seventeen two-color frieze groups

These are exactly the seventeen different patterns that the Symmetry Tribe illustrated, using a mouse motif of course, on the frieze tabletops:

The different mouse patterns are given in the same order as the footprint patterns above, with slight changes in the glide reflections for some of the patterns.