World-structure and non-Euclidean honeycombs

摘要: Milne (1934) described a one-dimensional system of discrete particles in uniform relative motion such that the aspect whole is same from each particle. The purpose present paper to construct analogous systems two and three dimensions. If uniformly moving observers regraduate their clocks so as describe other relatively stationary, private Euclidean spaces Special Theory Relativity become public hyperbolic space. This point view leads discussion honeycombs space, four which were discovered by Schlegel (1883, p. 444). One new honeycombs, called {4, 4, 3}, has for its vertices points whose co-ordinates are proportional integral solutions Diophantine equation t 2 - x y z = 1. As by-product, simple set generators generating relations obtained group all Lorentz transformations (Schild 1949, 39). Another by-product enumeration those groups generated reflexions space fundamental regions tetrahedra finite volume. work culminates discovery point-distribution mesh seven times close though apparently still far too coarse be direct cosmological significance. It follows some irregularity distribution extragalactic nebulae almost certainly geometrically inevitable.