Orthogonal hyperspheres

Abstract

Umbilical projection ([12], [14]) is a process suggested to derive results rather quickly in regard to four intersecting spheres [17] andn+1 intersecting hyperspheres in ann-space [18]. The same has been used with an advantage to deduce a porism on 2n+5 hyperspheres in ann-space [23]. The purpose of this paper is to concentrate on mutually orthogonal hyperspheres only and to illustrate simultaneously once again the utility and facility of this tool to arrive at a number of new and interesting results as follows:

The 2(n+1) intersections ofn+1 mutually orthogonal hyperspheres in ann-space, takenn at a time, give rise to 2ⁿ pairs ofsemi-inverse [22] simplexes, perspective from their radical centreH, such that the 2ⁿ primes of perspectivity coincide with their 2ⁿ hyperplanes of similitude and form anS-configuration (S-C) [15] with theircentral simplex S(A) as itsdiagonal simplex. Everysimplex of intersection introduced here isisodynamic [25] such that itstangential simplex, circumscribed to it along circumhypersphere, is perspective to it from itsLemoine point L. ItsLemoine hyperplane l, as the polar prime ofL w. r. t. it, is the same as that of itscomplementary simplex of intersection and coincides whith their prime of perspectivity such that their 2(n+1) altitudes are met by their commonBrocard diameter through their Lemoine points. The 2ⁿ Brocard diameters of the 2ⁿ pairs of complementary simplexes of intersection concur atH. The$\left( \begin{gathered} n + 1 \hfill \\ 2 \hfill \\ \end{gathered} \right)$ hyperspheres of antisimilitude of the given hyperspheres, having centres in a prime of similitude, form the commonNeuberg hyperspheres of the pair of semi-inverse simplexes, having this prime as their common Lemoine hyperplane, are consequently orthogonal to their cirumhyperspheres whose radical hyperplane, too, coincides whith this prime, and therefore belong to acoaxal net [15] passing through the pair of their commonNeuberg points on their common Brocard diameter. The second centres of similitude of the 2ⁿ pairs ofcomplementary hyperspheres of intersection form the 2ⁿ vertices of the dual [15] of the (S-C), whithS(A) as common diagonal simplex, as its polar reciprocal w. r. t. the common orthogonal hypersphere of then+1 hyperspheres, the first centres of similitude coinciding atH.

Due inspiration is derived from the works ofCourt ([2]–[9]) on mutually orthogonal circles and spheres.