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The packing of certain sets of cubes

Published online by Cambridge University Press:  24 October 2008

S. W. P. Steen
Affiliation:
Christ's College

Extract

Let (X1, …, Xn) be the coordinates of the centre of a unit cube C, in n-dimensional space, whose (n − 1)-dimensional faces are parallel to the axes of coordinates. Further let the X's be integers. Let εi = ± 1 for r (≤ n) values of i, 1 ≤ in, and let εi = 0 for the remaining nr values of i. Then (X1 + ε1, …, Xn + εn) gives the centre of a cube C′, which touches the cube C along an (nr)-dimensional edge or face. The cube C and the cubes C′, for all possible arrangements of the ε's, which are subject to the above conditions, form a symmetrical arrangement of cubes. This paper discusses the possibility of completely filling space by means of the packing together of such sets of cubes.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1931

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