page_title: Isohedral figure
hierarchical_section_title: Isohedral figure
caption_attribution_description: English: A regular tetrahedron, a cube, a regular octahedron, a pentagonal trapezohedron, a regular dodecahedron, a regular icosahedron, a tetrakis cube, a rhombic triacontahedron and a triakis icosahedron.
context_page_description: In geometry, a polytope of dimension 3 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.
  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 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.
  A polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral. They are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively.
context_section_description: 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.
  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.
  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.
  Note: not all isozonohedra are isohedral. Example: a rhombic icosahedron is an isozonohedron but not an isohedron.
caption: A set of isohedral dice
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