Conjugate

A conjugate of a polytope $$P$$ with coordinates in a field $$F $$ is the polytope $$\overline{P}$$created when an automorphism of $$F $$ is applied to the coordinates of the vertices of $$P$$. The number of conjugates of a polytope depends on the number of automorphisms of F (see Choice of field).

Properties
Conjugates do not depend on the position, size, or orientation of the original polytope. The resulting conjugate may also be transformed, but its shape does not change. A brief proof follows: Additionally, conjugate polytopes always have the same symmetries.
 * Translating a polytope by a vector $$\mathbf{v}$$ translates the conjugate polytope by its component-wise conjugate $$\overline{\mathbf{v}}$$, since field automorphisms respect addition.
 * Likewise, scaling a polytope by a factor $$s$$ scales the conjugate polytope by its conjugate $$\overline{s}$$, since field automorphisms respect multiplication.
 * Rotating and/or reflecting a polytope is equivalent to multiplying its coordinates by a matrix $$Q$$ such that $$Q^TQ=I$$. The coordinates of the conjugate polytope, then, are multiplied by its element-wise conjugate $$\overline{Q}$$. Since field automorphisms respect matrix multiplication, $$\overline{Q}^T\overline{Q} = \overline{I}$$, and since automorphisms also preserve the identity matrix, $$\overline{Q}^T\overline{Q} = I$$. Thus $$\overline{Q}$$ also represents a rotation and/or reflection of the conjugate polytope.

Choice of field
Polytopes are often defined with real coordinates, but $$\mathbb{R}$$ has only the trivial automorphism, which does not allow for non-trivial conjugates. Instead, the coordinate field is restricted to an algebraic field, e.g. $$\mathbb{Q}(\sqrt{5})$$, which may have non-trivial automorphisms. It appears that each polytope $$P$$ has a "canonical field" which is just large enough to represent the coordinates of some position, size, and orientation of $$P$$, and whose automorphisms create all conjugates of $$P$$. However, not all automorphisms of this canonical field may create real conjugate polytopes, as they may output complex numbers.

Examples

 * The rational numbers have no automorphisms besides the identity, so all polytopes which may be written with rational coordinates, such as oct and ico, have no non-trivial conjugates.
 * Regular polygons and polygon compounds with the same number of sides and connected components (e.g. heg, hag, and gahg) are all conjugates.
 * Snid, gosid, gisid, and girsid are conjugate polyhedra whose coordinates lie in a sextic field, four of whose automorphisms preserve real numbers.