If a polytope is realized such that its vertices lie at coordinates in a field, a conjugate polytope is formed by transforming the vertices using an automorphism of the field. The resulting polytope is combinatorially identical. Although commonly viewed as an operation, it is slightly more accurately viewed as an equivalence relation; a polytope may have multiple conjugates, or may not have any conjugates other than itself.
The field in question is usually chosen to be the field of algebraic numbers , whose realization admits many common polytopes defined by symmetry. For example, many convex regular polygons have conjugates in the form of regular star polygons; in particular, the regular heptagon, the heptagram, and the great heptagram are all conjugates. The conjugates of regular-faced polyhedra are the results of symmetrically replacing faces with conjugates; for example, the small stellated dodecahedron and great dodecahedron are conjugates.
The conjugates of uniform polytopes are always uniform, and the conjugates of scaliform polytopes are always scaliform. Because of this, conjugates can be a useful tool to find new polytopes with such properties.
Definition[edit | edit source]
A conjugate of a polytope with coordinates in a field is the polytope created when an automorphism of is applied to the coordinates of the vertices of . The number of conjugates of a polytope depends on the number of automorphisms of (see Choice of field).
Algebraic conjugates[edit | edit source]
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 that's the root of some polynomial with rational coefficients. That is to say, there exist rational such that
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[edit | edit source]
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:
- Translating a polytope by a vector translates the conjugate polytope by its component-wise conjugate , since field automorphisms respect addition.
- Likewise, scaling a polytope by a factor scales the conjugate polytope by its conjugate , since field automorphisms respect multiplication.
- Rotating and/or reflecting a polytope is equivalent to multiplying its coordinates by a matrix such that . The coordinates of the conjugate polytope, then, are multiplied by its element-wise conjugate . Since field automorphisms respect matrix multiplication, , and since automorphisms also preserve the identity matrix, . Thus also represents a rotation and/or reflection of the conjugate polytope.
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.
Choice of field[edit | edit source]
Polytopes are often defined with real coordinates, but 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. , which may have non-trivial automorphisms.
It appears that many polytopes have a "canonical field" , which is just large enough to represent the coordinates of some position, size, and orientation of , and whose automorphisms create all conjugates of . However, some polytopes require additional dimensions to use the canonical field. For example, the canonical field of the pentagon is , but it cannot be represented in and must be embedded in 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[edit | edit source]
- 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 polygons 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 and whose canonical fields share no automorphisms are the prism products of the conjugate(s) of and . For example, the 5-8, 5-8/3, 5/2-8, and 5/2-8/3 duoprisms are conjugates. However, if and 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.
References[edit | edit source]
- Makoto Kato (2013). "Is an automorphism of the field of real numbers the identity map?".