If S is a given regular d-simplex of edge length a in the d-dimensional Euclidean space $$\mathcal {E}$$
 , then the distances $$t_1$$
 , $$\ldots $$
 , $$t_{d+1}$$
 of an arbitrary point in $$\mathcal {E}$$
 to the vertices of S are related by the elegant relation $$\begin{aligned} (d+1)\left( a^4+t_1^4+\cdots +t_{d+1}^4\right) =\left( a^2+t_1^2+\cdots +t_{d+1}^2\right) ^2. \end{aligned}$$
 The purpose of this paper is to prove that this is essentially the only relation that exists among $$t_1,\ldots ,t_{d+1}.$$
 The proof uses tools from analysis, algebra, and geometry.

Distances from the Vertices of a Regular Simplex

The investigation of the relation among the distances of an arbitrary point in the Euclidean space $$\mathbb {R}^n$$
 to the vertices of a regular n-simplex in that space has led us to the study of simplices having a regular facet. Calling an n-simplex with a regular facet an n-pre-kite, we investigate, in the spirit of Hajja and Walker (Int J Math Ed Sci Tech 32:501–508, 2001), Edmonds et al. (Beitr Algebra Geom 46:491–512, 2005b), Edmonds et al. (Results Math 47:266–295, 2005a), and Hajja (Results Math 49:237–263, 2006), and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if $$n \ge 3$$
 , then the intersection of the family of n-pre-kites with any of the four known special families is the family of n-kites, thus extending the result in Hajja et al. (Beitr Algebra Geom 56:269–277, 2015). A basic tool is a closed form of a determinant that arises in the context of a certain Cayley–Menger determinant, and that generalizes several determinants that appear in Edmonds et al. (Results Math 47:266–295, 2005a), Hajja (Results Math 49:237–263, 2006), and Hajja and Hayajneh (J Geom 105:539–560, 2014). Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.

Pre-kites: simplices having a regular facet

Proposition 5 of Book I of Euclid’s Elements, better known as the Pons Asinorum or the Bridge of Asses, and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. A satisfactory generalization of this statement to general tetrahedra and a strong generalization to orthocentric tetrahedra are obtained in Hajja (J Geom 93:71–82, 2009a), and generalizations to orthocentric d-simplices, d ≥ 3, are obtained in Hajja (Stud Sci Math Hungarica 46:263–273, 2009b). In this paper, the strong generalization is seen to hold for a fairly large family of tetrahedra that are referred to as distinguished. These include acute and rectangular orthocentric tetrahedra as well as circumscriptible and isogonic tetrahedra. It is also seen that the strong generalization fails for isodynamic tetrahedra and that even a weak generalization fails for simplices in dimension four.

The pons asinorum and related theorems for tetrahedra

A theorem of D. Pompeiu states that the distances from the vertices of an equilateral triangle to any point in its plane can serve as the side lengths of a triangle. In this paper, it is proved tha...

Another n-Dimensional Generalization of Pompeiu’s Theorem

In the d dimensional Euclidean space, any set of n+1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and of radius R, with n positive and less than d+1, and a n flat when n is positive and less than d. The projection of O on the n flat is named O'. The focus is set on circumspheres which are contained in this unit ball. For any d larger than 1 and any n positive and less than d, the joint probability density function of the distance D = O'C and of R has a simple closed form expression. Their marginal probability density functions are both products of powers and of a Gauss hypergeometric function. Stochastic representations of D and R are expressed as geometric means of two independent beta random variables. For n equal to d, positive, D and R have a Dirichlet distribution while D and R are each beta distributed. Results of Monte-Carlo simulations are in very good agreement with their calculated counterparts

John Bentin

Papers

79.15 Regular Simplicial Distances

81.32 Regular polygonal distances