FRIEZE PATTERNS FOR REGULAR STAR POLYTOPES AND STATISTICAL HONEYCOMBS

Abstract

Much has been written about the discovery by physicists of quasicrystals and the almost simultaneous discovery by geometers of nonperiodic tessellations and honeycombs. A somewhat similar serendipity occurred when crystallographers saw what happens when identical balls of plastic clay or lead shot are shaken together and uniformly compressed, or when the bubbles (all of the same size) in a froth are measured; and almost simultaneously geometers investigated statistical honeycombs. Alternate doses of oil and water in a thin tube may be regarded as a one-dimensional "froth" {∞}, each "bubble" having just 2 neighbours. Analogously, soapsuds sandwiched between parallel glass plates (close together) may be regarded as a two-dimensional froth {6,3}, each bubble having just 6 neighbours. Three-dimensional froth presents a far more difficult problem because there is no regular honeycomb having 4 cells at each vertex. The best available substitute seems to be a "statistical" honeycomb {p, 3,3} where p, instead of being rational, is a real number such as π/ arctan \(\sqrt {\frac{1}{2}} \), somewhere between 5 and 6. ({5,3,3} is the regular 120-cell and {6,3,3} is non-Euclidean.) In such a statistical honeycomb, the number of neighbours for each bubble is 13.4, in good agreement with experiments in which the actual number is 12 or 14 and sometimes 15, but most often 13. Hoping not to be too fanciful, we venture to look for a statistical honeycomb {q,3,3,3} in Euclidean 4-space, q being a real number such as π/ arctan \(\sqrt {\frac{3}{5}} \), somewhere between 4 and 5. ({4,3,3,3} is the 5-cube while {5,3,3,3} is non-Euclidean.) In this case the number of neighbours for one bubble in the 4-dimensional froth is computed to be about 28.