• Corpus ID: 18079488

# REGULAR HONEYCOMBS IN HYPERBOLIC SPACE

@inproceedings{Coxeter1956REGULARHI,
title={REGULAR HONEYCOMBS IN HYPERBOLIC SPACE},
author={H. S. M. Coxeter},
year={1956}
}
made a study of honeycombs whose cells are equal regular polytopes in spaces of positive, zero, and negative curvature. The spherical and Euclidean honeycombs had already been described by Schlaf li (1855), but the only earlier mention of the hyperbolic honeycombs was when Stringham (1880, pp. 7, 12, and errata) discarded them as "imaginary figures", or, for the two-dimensional case, when Klein (1879) used them in his work on automorphic functions. Interest in them was revived by Sommerville…

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## References

SHOWING 1-10 OF 13 REFERENCES
World-structure and non-Euclidean honeycombs
• Mathematics
Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
• 1950
Milne (1934) described a one-dimensional system of discrete particles in uniform relative motion such that the aspect of the whole system is the same from each particle. The purpose of the present
VI.—Division of Space by Congruent Triangles and Tetrahedra
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
Non-Euclidean Geometry
Euclid’s Elements is dull, long-winded, and does not make explicit the fact that two circles can intersect, that a circle has an outside and an inside, that triangles can be turned over, and other
Regular figures in w-dimensional space
• Amer. J. Math
The nine regular solids
• Proc. 1st Canadian Math. Congress
• 1946
On close-packings of spheres in spaces of constant curvature
• Pubi. Math. Debrecen
• 1953
Theorie der homogen zusammengesetzten Raumgebilde
• Nova Acta Leop. Carol