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Let $\mathbb{S}^2$ be the unit sphere in $\mathbb{R}^3$, and let $x_1,x_2,x_3,x_4 \in \mathbb{S}^2$.

Suppose that $\sum_i x_i=0$, where we sum the vectors in $\mathbb{R}^3$.

Question: Does one of the following two options hold?

  1. The $x_i$ form a regular tetrahedron.
  2. The $x_i$ lie on a great circle in $\mathbb{S}^2$. (In that case, they must in fact form a rectangle).

Motivation:

"Balanced configurations" $x_i \in \mathbb{S}^2$ satisfying $\sum_i x_i=0$, coincide with the maximizes of the the total squared distance energy $$ E(x_i)=\sum_{i < j}\| x_i - x_j \|^2, $$ where $\| x_i - x_j \|$ denotes the Euclidean distance in $\mathbb{R}^3$.

It turns out that the set of such balanced configurations form a $5$-dimensional submanifold of $(\mathbb{S}^2)^4$, as it is the inverse image of the regular value zero, of the map $(x_i) \to \sum_i x_i$.

Asaf Shachar
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    From $x_i$s that forms a regular tetrahedron, rotate $x_1$ and $x_2$ about the line joining their midpoint and the origin, and the sum of $x_i$s should not change. – peterwhy Apr 17 '22 at 21:39

1 Answers1

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Consider these four points:

$$ (0.8,0.6,0) \\ (0.8,-0.6,0) \\ (-0.8,0,0.6) \\ (-0.8,0,-0.6) $$

David K
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  • Thanks. Can you please elaborate on how did you come up with this example? Was there some process of producing it? (like in
    peterwhy's comment).     – Asaf Shachar Apr 17 '22 at 21:49
  • I took a rectangle and rotated two adjacent vertices around the axis through their midpoint. – David K Apr 17 '22 at 22:45