Regular Honeycombs in Hyperbolic Space

made a study of honeycombs whose cells are equal regular polytopes in spaces of positive, zero, and negative curvature. The spherical and Euclidean honeycombs had already been described by Schlaf li (1855), but the only earlier mention of the hyperbolic honeycombs was when Stringham (1880, pp. 7, 12, and errata) discarded them as "imaginary figures", or, for the two-dimensional case, when Klein (1879) used them in his work on automorphic functions. Interest in them was revived by Sommerville (1923), who investigated their metrical properties. The honeycombs considered by the above authors have finite cells and finite vertex figures. It seems desirable to make a slight extension so as to allow infinite cells, and infinitely many cells at a vertex, because of applications to indefinite quadratic forms (Coxeter and Whitrow 1950, pp. 424, 428) and to the close packing of spheres (Fejes Tóth 1953, p. 159). However, we shall restrict consideration to cases where the fundamental region of the symmetry group has a finite content, like that of a space group in crystallography. This extension increases the number of three-dimensional honeycombs from four to fifteen, the number of four-dimensional honeycombs from five to seven, and the number of five-dimensional honeycombs from zero to five. A further extension allows the cell or vertex figure to be a star-poly tope, so that the honeycomb covers the space several times. Some progress in this direction was made in two earlier papers: one (Coxeter 1933), not insisting on finite fundamental regions, was somewhat lacking in rigour; the other (Coxeter 1946) was restricted to two dimensions. The present treatment is analogous to § 14.8 of Regular Polytopes (Coxeter 1948, p. 283). We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane. 2. Two-dimensional honeycombs. In the Euclidean plane, the angle of a regular ^>-gon, {p}, is (1 — 2\p)n. In the hyperbolic plane it is smaller, gradually decreasing to zero when the side increases from 0 to oo. Hence, if p and q are positive integers satsifying