Abstract: Let ( X 1 , …, X n ) 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 ε i = ± 1 for r (≤ n ) values of i , 1 ≤ i ≤ n , and let ε i = 0 for the remaining n − r values of i. Then ( X 1 + ε 1 , …, X n + ε n ) gives the centre of a cube C ′, which touches the cube C along an ( n − r )-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.
Publication Year: 1931
Publication Date: 1931-01-01
Language: en
Type: article
Indexed In: ['crossref']
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