Shapes where all points are of equal distance

Question

I'm interested in shapes where all points are equal distance from one another, like an equalateral triangle.

A square not being examples of this, because the opposing corner vertices are not the same distance from one another as the sides.

Please can somebody point me in the direction of any higher dimensional analysis of this that might exist. If we increase the number of dimensions, can we find shapes where all points are equal distance?

Thank you,

Steven

Answer 1

Here is simple example. Consider $\mathbb{R}^n$ and the $n$ points $e_1 = (1,0,\dots,0)$, $e_2 = (0,1,0,\dots,0)$, ..., $e_n = (0,\dots,0,1)$. It is straightforward to check that all points have the same distance to each other. The convex hull of these points is known as the standard/unit $(n-1)$-simplex, as it forms a $n-1$ dimensional manifold.

• The standard 0-simplex is a point at $(1)$.
• The standard 1-simplex is a line segment between $(1,0)$ and $(0,1)$.
• The standard 2-simplex is a equilateral triangle with vertices $(1,0,0)$, $(0,1,0)$, $(0,0,1)$.
• The standard 3-simplex is a regular tetrahedron.
• etc...