World-Structure and Non-Euclidean Honeycombs
Abstract
Milne (1934) described a one-dimensional system of discrete particles in uniform relative motion such that the aspect of the whole system is the same from each particle. The purpose of the present paper is to construct analogous systems in two and three dimensions. If the uniformly moving observers regraduate their clocks so as to describe each other as relatively stationary, the private Euclidean spaces of the Special Theory of Relativity become public hyperbolic space. This point of view leads to a discussion of uniform honeycombs in hyperbolic space, four of which were discovered by Schlegel (1883, p. 444). One of the new honeycombs, called {4, 4, 3}, has for its vertices the points whose four co-ordinates are proportional to the integral solutions of the Diophantine equation t2-x2-y2-z2 = 1. As a by-product, a simple set of generators and generating relations are obtained for the group of all integral Lorentz transformations (Schild 1949, p. 39). Another by-product is the enumeration of those groups generated by reflexions in hyperbolic space whose fundamental regions are tetrahedra of finite volume. The work culminates in the discovery of a point-distribution whose mesh is seven times as close as that of {4, 4, 3}, though apparently still far too coarse to be of direct cosmological significance. It follows that some irregularity in the distribution of the extragalactic nebulae is almost certainly geometrically inevitable.
- Publication:
-
Proceedings of the Royal Society of London Series A
- Pub Date:
- April 1950
- DOI:
- 10.1098/rspa.1950.0070
- Bibcode:
- 1950RSPSA.201..417C