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

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Springer International Publishing
Copyright © 2017 by Springer International Publishing
Mathematics; Mathematics, general
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