Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T17:02:49.826Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.