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A cube complex, in the cubical CW complex convention used here, is built from standard hypercubes $[0,1{]}^n$ by identifying their faces of a cube using isometries. The restriction of each characteristic map to a face of a cube agrees with the characteristic map of a lower-dimensional cell, after an isometry of that face. The cubes carry their Euclidean metrics, giving an induced length distance on each connected component.^[ref]

Face identifications may identify different boundary points of the same cube. For example, one interval with its endpoints identified is a circle, and one square with opposite sides identified is a torus. The latter has one vertex, two edges and one square; it is the Salvetti complex for two commuting generators.

A stricter polyhedral convention requires every closed cube to embed and every pair of cubes to meet in one common face. That convention needs a finer cell decomposition of these examples. The cubical-CW convention allows the one-vertex quotient models used in geometric group theory. Being a nonpositively curved cube complex is an additional condition, detected by the link of a vertex in a cube complex.

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.

For $\mathbf{b}\ne 0$, define the orthogonal projection of $\mathbf{a}$ onto its span byThe first is parallel or antiparallel to $\mathbf{b}$, or zero, and ${\mathbf{a}}_{\perp }\cdot \mathbf{b}=0$. This proves the decomposition. If $\mathbf{a}\ne 0$ as well, comparison with ${\mathbf{a}}_{\parallel }=|\mathbf{a}|\cos \theta \hat{\mathbf{b}}$ yieldsIndeed $|\mathbf{a}{|}^2=|{\mathbf{a}}_{\parallel }{|}^2+|{\mathbf{a}}_{\perp }{|}^2$ shows that the ratio lies in $[-1,1]$. There is a unique $\theta \in [0,\pi ]$ with this cosine. The definition is symmetric in the two vectors, unchanged by orthogonal coordinate changes and positive rescaling, and agrees with ordinary plane geometry in their span: projecting a unit vector onto the other gives its cosine. Parallel, perpendicular and antiparallel directions give $0,\pi /2,\pi$ respectively. The zero vector has no defined angle; if $\mathbf{b}=0$, the nonzero-axis projection formula itself must not be used.

The vertices of the centered hypercube are sign vectors $\mathbf{u}\in \{-1,1{\}}^n$. A body diagonal identifies $\mathbf{u}$ and $-\mathbf{u}$, hence there are $2^{n-1}$ such diagonals. If two sign vectors disagree in $h$ positions, their dot product is $n-h-h=n-2h$. For odd $n$ this integer is odd and cannot be zero. Thus no two body diagonals are perpendicular in odd dimension, by the orthogonality of hypercube body diagonals.

In four dimensions, mutually orthogonal nonzero vectors are linearly independent, so at most four mutually perpendicular diagonals are possible. Four are achieved by the directionsEach has squared norm four and every distinct pair has zero dot product; these are the rows of a Hadamard matrix. Therefore the maximum is four.

For distinct diagonals, the Hamming distance is $h=1,2,3$; $h=0,4$ would represent the same line. Oriented vertex directions therefore give cosines $1/2,0,-1/2$. The angles areFor unoriented diagonals, the usual smaller angle between the lines is $\pi /3$ or $\pi /2$; the supplementary intersection angle $2\pi /3$ also occurs when the two rays are chosen oppositely. Stating this convention avoids counting a line and its reverse as different diagonals.

A signed permutation on $n$ labels is a permutation of the symbols $\{\pm 1,\dots ,\pm n\}$ satisfying $w(-i)=-w(i)$. It is specified by the list $w(1),\dots ,w(n)$, whose absolute values form a permutation of $1,\dots ,n$. For example, $(-2,3,-1)$ specifies $w(1)=-2$, $w(2)=3$, $w(3)=-1$. There are $2^{n}n!$ choices: select an ordinary permutation and an independent sign for each entry. Under composition they form the hyperoctahedral group, acting on a hypercube by permuting coordinate axes and reversing their signs.