# Isohedral figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is **isohedral** or **face-transitive** when all its faces are the same. More specifically, all faces must be not merely congruent but must be *transitive*, i.e. must lie within the same *symmetry orbit*. In other words, for any faces *A* and *B*, there must be a symmetry of the *entire* solid by rotations and reflections that maps *A* onto *B*. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.^{[1]}

Isohedral polyhedra are called **isohedra**. They can be described by their face configuration. A form that is isohedral and has regular vertices is also edge-transitive (isotoxal) and is said to be a quasiregular dual: some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted. An isohedron has an even number of faces.^{[2]}

A polyhedron which is isohedral has a dual polyhedron that is vertex-transitive (isogonal). The Catalan solids, the bipyramids and the trapezohedra are all isohedral. They are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, and face-transitive (isogonal, isotoxal, and isohedral). A polyhedron which is isohedral and isogonal is said to be noble.

Not all isozonohedra^{[3]} are isohedral.^{[4]} Example: a rhombic icosahedron is an isozonohedron but not an isohedron.^{[5]}

A polyhedron (or polytope in general) is ** k-isohedral** if it contains

*k*faces within its symmetry fundamental domain.

^{[6]}

Similarly a ** k-isohedral tiling** has

*k*separate symmetry orbits (and may contain

*m*different shaped faces for some

*m*<

*k*).

^{[7]}

A **monohedral** polyhedron or monohedral tiling (*m* = 1) has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An ** r-hedral** polyhedra or tiling has

*r*types of faces (also called dihedral, trihedral for 2 or 3 respectively).

^{[8]}

Here are some example k-isohedral polyhedra and tilings, with their faces colored by their *k* symmetry positions:

A **cell-transitive** or **isochoric** figure is an *n*-polytope (*n* > 3) or honeycomb that has its cells congruent and transitive with each other. In 3-dimensional honeycombs, the catoptric honeycombs, duals to the uniform honeycombs are isochoric. In 4-dimensions, isochoric polytopes have been enumerated up to 20 cells.^{[9]}

A **facet-transitive** or **isotopic** figure is a *n*-dimensional polytopes or honeycomb, with its facets ((*n*−1)-faces) congruent and transitive. The dual of an *isotope* is an isogonal polytope. By definition, this isotopic property is common to the duals of the uniform polytopes.