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The shape is made of nothing but equilateral triangles, and the points obviously for each one are equidistant, and just thinking about it there are several axes upon which the shape has rotational symmetry for three flips. Four, I think.

Also, as an aside, if there is a generalized version of this for higher dimensions I'd be interested to see it. My guess is that a regular tetrahedron is the shape that arises when four points are equidistant, and I suspect it is not possible to add a fifth point and have all five equidistant in three dimensions. But this is all intuitive. How do I verify this?

Thanks!