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Space-filling Tetrahedra in Euclidean Space

Published online by Cambridge University Press:  20 January 2009

D. M. Y. Sommerville
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In the answer to the book-work question, set in a recent examination to investigate the volume of a pyramid, one candidate stated that the three tetrahedra into which a triangular prism can be divided are congruent, instead of only equal in volume. It was an interesting question to determine the conditions in order that the three tetrahedra should be congruent, and this led to the wider problem – to determine what tetrahedra can fill up space by repetitions. An exhaustive examination of this required one to keep an open mind as regards whether space is euclidean, elliptic, or hyperbolic, and then to pick out the forms which exist in euclidean space.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , February 1922 , pp. 49 - 57
Copyright
Copyright © Edinburgh Mathematical Society 1922

References

* See the author's paper: “Division of space by congruent triangles and tetrahedra,” Edinburgh, Proc. Soy. Soc., vol. 43 (1923), pp. 85–116.Google Scholar

* The necesaary and sufficient condition that a tetrahedron should be symmetrical is that a pair of adjacent edges should be equal, and the two edges opposite to them also equal.