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arXiv · 1609.06552

Distances from the vertices of a regular simplex

Mowaffaq Hajja·Mostafa Hayajneh·Bach Nguyen·Shadi Shaqaqha
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Abstract

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

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Mowaffaq Hajja, Mostafa Hayajneh, Bach Nguyen, Shadi Shaqaqha. 2016-09-21. Distances from the vertices of a regular simplex. https://arxiv.org/abs/1609.06552

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