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On the Orthogonal Groups in Four Dimensions

Published online by Cambridge University Press:  24 October 2008

G. de B. Robinson
G. de B. Robinson
Affiliation:
St John's College.
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Extract

Configurations of points in higher space have been known and studied for some time. In what follows we shall classify those configurations of points lying on the hyper-sphere in [4] which may be said to possess the group property. In this classification we shall seek to enumerate only those configurations which are essentially different from this point of view, in order to bring out any geometrical differences which might throw light on the fundamental problems of the theory of linear groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 1 , January 1931 , pp. 37 - 48
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* For references see Coxeter, , Phil. Trans. Royal Soc., A, 229, pp. 329–425.CrossRefGoogle Scholar

† E.g. all the semi-regular polytopes which were derived from the regular ones by Mrs Stott are essentially the same, and could be obtained by varying the position of the initial point. Cf. Journ. London Math. Soc. 6 (1931).Google Scholar

‡ E.g. that of Primitivity: cf. Blichfeldt, , Finite Coll. Groups, Chicago, 1917.Google Scholar

§ Goursat, , Annales de l'École Normale, (3), 6 (1889), p. 80Google Scholar. Goursat remarks: “Je ne m'occuperai pas dans ce travail de la recherche de toutes les divisions qui correspondent à tons les groupes qui viennent d'ètre énumérés.”

* Cf. Ford, , Automorphic Functions, New York, 1930, Chap. vi.Google Scholar

* Goursat, loc. cit., pp. 62–79.

† Bieberbach, , Math. Ann. 70 (1911), pp. 297–336.CrossRefGoogle Scholar

* An argument similar to that of § 2 could be applied to these 3-dimensional groups. The sub-group g would be cyclical.

* Cf. Klein, , Vorlesungen über das Ikosaeder (1884), Chap. 11, § 8.Google Scholar

* With this group Goursat associates δ1.

† Cf. Proc. Camb. Phil. Soc. 26 (1930), pp. 94–98.Google Scholar

* These two sets of coordinates for a 24-cell, or {3, 4, 3}, are given by Coxeter (loc. cit.), p. 347, except for a constant factor.

* Again, the sets of coordinates of ε4 and ε4′ are the same as those given by Coxeter, except for a constant factor.

* Cf. Proc. Camb. Phil. Soc. 26 (1930), p. 310.Google Scholar

† If the cycle be composed of spherical 2-cells, k = Γ/2, since the only transformation which can leave such a space unaltered is a reflection.

* A particularly simple reducible group of this type would generate the generalised prism [{k}, {k′}], if the initial point were properly chosen. Cf. Coxeter, loc. cit., p. 352.