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Spherical Simplexes in n-dimensions

Published online by Cambridge University Press:  20 January 2009

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All points in an n-space equidistant from a fixed point (the centre) constitute what may be called a spherical continuum of the nth order,—the continuum being of n — 1 dimensions ((n — l)-dimensional spread) and of the 2nd degree. Any region of this spherical continuum bounded by n (n — l)-dimensional linear continua or, primes (spaces of n — 1 dimensions), passing through the centre shall be called a spherical simplex of the nth order. This spherical simplex is bounded by n faces, spherical simplexes of the (n — l)th order, each of which in turn is bounded by n — 1 spherical simplexes of the (n — 2)th order, and so on till we reach spherical triangles, arcs and lastly points, the vertices. The total number of spherical simplexes of different orders connected with one of the nth order is 2n — 2. The n spherical continua of the (n — l)th order which contain the faces of the spherical simplex of the nth order determine a set of 2n spherical simplexes of the same order, 2n–1 pairs, the two spherical simplexes of a pair being symmetrically situated with respect to the centre and therefore congruent.

Research Article
Copyright © Edinburgh Mathematical Society 1930


page 6 note 1 The subject has been studied by Schläfli in the second chapter of his book Theorie der vielfachen Kontinuitat. The field of his investigation is different from that of the present paper and the method adopted is totally different. Also xbid. Coolidge, Non Euclidean Geometry.Google Scholar

page 7 note 1 This is in accordance with the nomenclature used by Prof, von Staudt, who has called the function the sine of the solid angle that the spherical triangle subtends at the centre of the sphere. Orelle 24 (1842), 252.Google Scholar

page 8 note 1 Schläfli, , loc. cit §20 (4), who has given only one of these forms. The formula is also proved by the consideration of a parallelochesm of the nth order whose content is equal to the product of the contents of two of its adjacent faces (parallelochesras of (n – 1)th order) multiplied by the sine of the angle between the faces and divided by the content of the parallelochesm of (n – 2)th order in which the faces intersect.Google Scholar

page 10 note 1 Owing to the importance of formula (13) we append another proof by the method of projection: Let a be the projection of n on P(Nn), b and c the projections of α on P(N nn–1 n–2), e the projection of c on bd and f that of a on ce.