A Geometric Higher Dimensional Phenomenon?

Let a $n$ cube have vertex coordinates in binary form for all $k=0$ to ${2}^{n}-1$ , such as for example for a square, $\{(0,0),(0,1),(1,0),(1,1)\}$ . A square is a $2$ -hypercube, where $n=2$ .

Let ${H}_{p}$ be a square matrix such that

${H}_{1}=\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{pmatrix}$

${H}_{p+1}=\begin{pmatrix} { H }_{ p } & { H }_{ p } \\ { H }_{ p } & { I }_{ p }-{ H }_{ p } \end{pmatrix}$

where ${I}_{p}$ is a square matrix of the same size as the ${H}_{p}$ but with all elements $1$ .

Then, each ${H}_{p}$ matrix contains the coordinates of the $n+1={2}^{p+1}$ vertices of a $n$ simplex sharing the same vertices as the $n$ hypercube, which are the $n+1$ rows with the first column left off.

The Pythagorean distance between any two vertices of the $n$ simplex as described by the ${H}_{p}$ matrix are all the same, here being $\sqrt{{2}^{p}}$ , which is a property of regular simplexes.

Since this will generate an infinite sequence of such ${H}_{p}$ square matrices, there’s an infinity of hypercubes with this property. The ${H}_{p}$ square matrices belong to a special class called Hadamard Matrices.

This is a simplified proof. A fuller proof takes longer. Below contains the coordinates for the $7$ -Simplex.

${H}_{2}=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \end{pmatrix}$