Conjugate

A conjugate of a polytope is created by changing the coordinates of another one in a specific manner, usually by flipping signs of roots. Conjugate polytopes are isomorphic as abstract polytopes to one another, though not all isomorphic polytopes are conjugate.

The corresponding facets of conjugate polytopes are connected in the same way, even if the polytopes may look different at first glance. Polygons in the polytope are typically exchanged for starry or non-starry versions of themselves, e.g. octagons become octagrams and vice versa in the conjugate.

In contrast to most other constructions on polytopes, conjugates depend on the exact representation of their coordinates. For example, a pentagon with vertices on the unit circle and including the vertex (1, 0) will have a pentagram as a conjugate, but a pentagon with edge length π will have no conjugates.

Conjugates of scaliform polytopes are always scaliform. Because of this, conjugates can be a useful tool to find new polytopes.

Definition
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).

Algebraic conjugates
To understand the effect conjugation has on the coordinates of a polytope, we need to look into a related mathematical notion.

An algebraic number is any real number $$\xi$$ that's the root of some polynomial with rational coefficients. That is to say, there exist rational $$a_0,\ldots,a_n$$ such that
 * $$a_0+a_1\xi+a_2\xi^2+\ldots+a_n\xi^n=0.$$

Every algebraic number has an associated minimal polynomial, which is the monic polynomial (its leading coefficient is 1) of least degree of which it is a root. Two algebraic numbers are said to be (algebraically) conjugate if they have the same minimal polynomial. Being conjugate is an equivalence relation.

It turns out that automorphisms of fields of real numbers must necessarily send numbers to algebraic conjugates, as they must satisfy the same algebraic equations involving rational numbers. In particular, this implies that the coordinates of conjugate polytopes are always algebraic conjugates of one another, hence the name. Note however that arbitrarily swapping coordinates by their algebraic conjugates won't necessarily yield a conjugate.

Properties
Conjugates do not depend on the position, size, or orientation of the original polytope, as long as these are changed within the same field. The resulting conjugate may also be transformed, but its shape does not change. A brief proof follows: Furthermore, conjugates of planar polytopes will also be planar, as field automorphisms respect bases for subspaces. Additionally, conjugate polytopes always have the same symmetries, and the same amounts of elements in each dimension. Some corresponding elements may not be exactly the same, but will be conjugates of each other.
 * 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. $$\mathbb{Q}\left[\sqrt{5}\right]$$, which may have non-trivial automorphisms.

It appears that many polytopes $$P$$ have a "canonical field" $$F$$, 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, some polytopes require additional dimensions to use the canonical field. For example, the canonical field of the pentagon is $$F=\mathbb{Q}(\sqrt{5})$$, but it cannot be represented in $$F^2$$ and must be embedded in $$F^3$$ instead (i.e. as a face of the dodecahedron). Additionally, not all automorphisms of the 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 the octahedron and icositetrachoron, have no non-trivial conjugates.
 * Regular polygon s with the same number of sides and connected components (e.g. the heptagon, heptagram, and great heptagram) 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.
 * The conjugates of a prism product of polytopes $$P$$ and $$Q$$ whose canonical fields share no automorphisms are the prism products of the conjugate(s) of $$P$$ and $$Q$$. For example, the 5-8, 5-8/3, 5/2-8, and 5/2-8/3 duoprisms are conjugates. However, if $$P$$ and $$Q$$ share a field, the set of conjugates becomes restricted, e.g. the 5-5 and 5/2-5/2 duoprisms are conjugates of each other, but not of the 5-5/2 duoprism.