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VI.—Division of Space by Congruent Triangles and Tetrahedra

Published online by Cambridge University Press:  15 September 2014

D. M. Y. Sommerville
D. M. Y. Sommerville
Affiliation:
Victoria University College, Wellington, N.Z.
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Extract

It is proposed to investigate the various ways in which it is possible to divide the plane into congruent triangles, and space of three dimensions into congruent tetrahedra. The method of inquiry will not at first reveal any distinction between elliptic, euclidean, and hyperbolic space; networks in all three spaces will be obtained concurrently, and they must be tested afterwards to determine the type of space to which they belong. In the first part we shall deal with the plane, and in the second part with space of three dimensions.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 43 , 1924 , pp. 85 - 116
Copyright
Copyright © Royal Society of Edinburgh 1924

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References

page 85 note * Throughout this paper we shall assume the antipodal form of elliptic space, i.e. spherical space; in the polar form some of the networks would not be possible without overlapping. Further, in hyperbolic space, as also in “euclidean, we shall assume that the vertices of the cells are proper points, not at infinity or ideal. Cf. Sansone, G., “Le divisioni regolari dello spazio iperbolico in piramidi e doppi piramidi,” Pisa, Ann. Scuola Norm., 13 (1919), pp. 129.Google Scholar

page 87 note * I have just seen the publisher's announcement of a book by Major MacMahon, P. A., entitled New Mathematical Pastimes (Cambridge Univ. Press, 1921)Google Scholar, which is partly devoted to the problem of fitting together congruent triangles, a form of what the author calls “generalised dominoes.”

page 101 note * In 4ii the four tetrahedra round the edge a′ form a regular octahedron. This honeycomb can therefore be derived from the regular 24-cell by drawing one diagonal of each of the octahedra. This can be done in a variety of ways. The most regular ones are (a) that in which two diagonals are at each node; at each node there are 8 vertices P, S and 8 vertices Q, R; and (b) that in which there are 6 diagonals at each of 8 nodes, and none at the remaining 16 nodes: at these 8 nodes there are 24 vertices Q, R, and at the remaining 16 there are 12 vertices P, S.