Hypercube ($[-1,1{]}^n$)

A hypercube is the Cartesian product of $n$ equal-length closed intervals in Euclidean space. The centered cube $[-1,1{]}^n$ has $2^n$ vertices, each with coordinates in $\{-1,1\}$. Antipodal vertices determine $2^{n-1}$ body diagonals.

Two sign-vector vertices of a centered hypercube differing in $h$ coordinates have dot product $n-2h$. Thus their body diagonals can be orthogonal only in even dimension. A collection of mutually orthogonal nonzero diagonals has at most $n$ members; achieving $n$ is equivalent to the existence of a Hadamard matrix of order $n$. For unoriented lines the angle is determined by $|n-2h|/n$, while oriented vertex vectors retain the sign of their dot product.