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Perspective simplexes †

Published online by Cambridge University Press:  09 April 2009

Sahib Ram Mandan
Sahib Ram Mandan
Affiliation:
Indian Institute of TechnologyKharagpur, India
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The main purpose of this paper is to prove the proposition: “A set of r mutually perspective (m.p.) (s—1)-simplexes have the same [s−2] (say x) of perspectivity, if and only if their centres of perspectivity (c.p.) lie in an [r−2] (say y); there then arises another such set of s m.p. (r—1)-simplexes, having the same rs vertices, which have y as their common [r−2] of perspectivity such that their c.p. lie in x.” The proposition is true in any [k] for k = s−1, s,…, r+s−2 (r ≦ s). The configuration of the proposition in [r+s−2] arises from the incidences of any r+s arbitrary primes therein and is therefore invariant under the symmetric group of permutations of r+s objects, and that in [r+s−3] is self-dual and therefore self- polar for a quadric therein. Some special cases of some interest for r = s are deduced. The treatment is an illustration of the elegance of the Möbius Barycentric Calculus ([15], pp. 136–143; [1], p. 71).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 1 , February 1966 , pp. 11 - 17
Copyright
Copyright © Australian Mathematical Society 1966

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