Dimension vector

The dimension vector of a polytope is a vector which classifies the symmetries of a polytope. Each value of the the dimension vector gives the dimension of a generating mirror of a polytope. Dimension vectors are most typically used in the description of regular skew polytopes, however they can be applied in some other situations.

Definition

Given an isometric involution X , its mirror is the set of fixed points of X . This mirror forms an affine plane and thus has a dimension.

The dimension vector of a rank n  polytope with generating mirrors $\langle \rho _{0},\dots ,\rho _{n}\rangle$ is a vector of size n  where the i th element is the dimension of the mirror $\rho _{n}$.

Choice of mirrors

The dimension vector depends on the choice of generating mirrors. If we are restricted to the case where the space that the generators acts on is the same as the linear span of the vertices (i.e. the dimension of the space is the same as the dimension of the polytope), then the dimension vector is unambgious. "The" dimension vector is always this vector unless otherwise noted. However if the dimension ambient space exceeds that of the linear span of the vertices (for example a flat hexagon in 3D space), there are several ways to choose mirrors of different dimensions. This is because the mirrors can be extended into the space not spanned by the vertices in any way without changing its action on the vertices and thus not changing the polytope.

• Example mirrors of a flat hexagon in 3-dimensional space
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  A regular hexagon with mirrors of dimension (1,1).

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  A regular hexagon with mirrors of dimension (1,2).

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  A regular hexagon with mirrors of dimension (2,1).

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  A regular hexagon with mirrors of dimension (2,2).

These sorts of generators can appear as the result of certain operations, for example the dual of the skew hexagon is the flat hexagon above with dimension vector (2,1).