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The groups of symmetries of the regular polytopes

Published online by Cambridge University Press:  24 October 2008

J. A. Todd
J. A. Todd
Affiliation:
Trinity College.
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Extract

The object of the present paper is the geometrical study of the groups of rotation and reflexion of the regular polytopes in higher space, and the extension to these configurations of known results in the cases of the ordinary regular polyhedra. It will appear from the work that the groups can be defined abstractly in terms of a certain number of operations, the relations connecting which have a particularly simple form. For a polytope in n dimensions this number is n − 1, if we consider the group composed simply of the rotations of the polytope, while if we consider the extended group, which includes the possible reflective symmetries of the polytope, n operations suffice. It appears, further, that with one exception all the groups so obtained possess the property that their operations are expressible in terms of two, so that the entire group can be generated by two operations. The relations connecting these, however, are in general complicated, and the symmetrical forms involving more operations are more convenient to use.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 2 , April 1931 , pp. 212 - 231
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Moore, , Proc. Lond. Math. Soc. (1) 28 (1897), 357–366.Google Scholar

* We shall only consider finite convex regular polytopes.

† Schläfli, , Journal de Math. 20 (1855), 361Google Scholar; see also Coxeter, , Phil. Trans. (A) 229 (1930), 342 et seq.Google Scholar

* Cf. Klein, , Vorlesungen über das Ikosaeder (1884), chap. I.Google Scholar

* I am indebted to Mr H. S. M. Coxeter for this proof.

* It will be convenient to define the relation U ij. U jk = U ik then holds without restriction.

* Cf. Coxeter, , loc. cit. ante, 378.Google Scholar

* Goursat, , Ann. École Normale Supérieure (3), 6 (1889), 80.Google Scholar

* Van, Oss, Verh. Akad. van Wet. Amsterdam, 7 (1899), 1.Google Scholar