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The densities of the regular polytopes, Part 3*

Published online by Cambridge University Press:  24 October 2008

H. S. M. Coxeter
H. S. M. Coxeter
Affiliation:
Trinity College
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Extract

This third and last part of the paper is concerned with the interpretation of the Schläfli symbol {k1, k2, …, km−1} when the k's are unrestricted. It is shown that, whenever the k's are integers (greater than 2), the symbol represents a polytope in a generalized space having time-like as well as space-like dimensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 1 , January 1933 , pp. 1 - 22
Copyright
Copyright © Cambridge Philosophical Society 1933

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References

† “The regular divisions of space of n dimensions and their metrical constants”, Rend. di Palermo, 48 (1923), 9–22.Google Scholar

* Lobatschewski, N. I., Zwei geometrische Abhandlungen (Teubner, 1898)Google Scholar. See also Weyl, H., Space—Time—Matter (Methuen, 1922), 77Google Scholar, and numerous textbooks on non-Euclidean geometry.

† Lorentz-Einstein-Minkowski, , Das Relativitätsprinzip (Teubner, 1920), 54Google Scholar. For an account in English, see Silberstein, L., Theory of Relativity (Macmillan, 1924), 127–140.Google Scholar

* Geometry of n dimensions (Methuen, 1929), 125.Google Scholar

* E.g. the ideal region of Lobatschewskian 4-space is de Sitter's world, while S 3T is Minkowski's world.

* Merely “solid”, if n=3.

* By spherical trigonometry,

* If the reader is unable to verify this assertion, he can use the following alternative argument. If (5 5 3) was infinite, ( 3 5 3) would have to be infinite too; but if (5 5 3) is finite, (5, 5, 3) must be divisible into a finite number of elementary simplexes. All the vertices of these must be actual, since all those of (5, 5, 3) are actual. If ( 3 5 3) was finite, a finite number of these elementary simplexes would fill ( 3, 5, 3); which is absurd, since ( 3, 5, 3) has an ideal vertex.

* The extra terms that can be appended when p = 5 and m < 5 now appear to be “freaks”. The actual vertex figures of do not generally suffice to bound a polytope.

* Alternatively,

since

* Cf. Coxeter, , Phil. Trans. Royal Soc. A 229 (1930), 357 (§ 5.3)CrossRefGoogle Scholar.

* “Theorie der vielfachen Kontinuität”, Neue Denkschr. d. allg. schweiz. Gesells. f. d. gesammten Naturwiss. 38 (1901), 98 (§ 29)Google Scholar. We have interchanged Schläfli's m, n.

† Minkowskian if m>1+sec (2π/p).

‡ To avoid repetition.