Clifford’s chain and its analogues in relation to the higher polytopes

Abstract

Clifford’s chain of theorems on circles in a plane, Grace’s sequence of theorems on spheres, and Grace’s and Brown’s sequence on hyperspheres in four dimensions are unified and completed. This is achieved by demonstrating a correspondence between each configuration and one of Coxeter’s polytopes p gr in space of ( p + g + r + 1) dimensions. Thus Clifford’s configurations of points and circles in a plane correspond to the polytopes 1 1 , r r'. Grace’s figures of points and spheres correspond to polytopes l 2 r ; and Grace’s and Brown’s figures of points and hyperspheres correspond to polytopes l 3 r . As a result it is shown that to Grace’s sequence there may be added two more symmetrical configurations, one of 17280 points and 240 spheres, the other of an infinite number of points and spheres, before the sequence terminates. To Grace’s and Brown’s sequence may be added just one more symmetrical configuration of points and hyperspheres. Furthermore, it is shown that in space either of five or of six dimensions there exists a finite sequence of three configurations; in any space of seven or more dimensions there is a sequence of just two configurations each. A related chain of theorems due to Homersham Cox, more general than Clifford’s, is likewise shown to have analogues in spaces of higher dimensions.