# Distances from the Vertices of a Regular Simplex

Distances from the Vertices of a Regular Simplex If S is a given regular d-simplex of edge length a in the d-dimensional Euclidean space $$\mathcal {E}$$ E , then the distances $$t_1$$ t 1 , $$\ldots$$ … , $$t_{d+1}$$ t d + 1 of an arbitrary point in $$\mathcal {E}$$ 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} ( d + 1 ) a 4 + t 1 4 + ⋯ + t d + 1 4 = a 2 + t 1 2 + ⋯ + t d + 1 2 2 . The purpose of this paper is to prove that this is essentially the only relation that exists among $$t_1,\ldots ,t_{d+1}.$$ t 1 , … , t d + 1 . The proof uses tools from analysis, algebra, and geometry. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# Distances from the Vertices of a Regular Simplex

, Volume 72 (2) – May 15, 2017
16 pages

/lp/springer_journal/distances-from-the-vertices-of-a-regular-simplex-H1rBBdLfMi
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0689-1
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Results in MathematicsSpringer Journals

Published: May 15, 2017

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