thumb|A set of isohedral dice 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.Grünbaum (1960)
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 are isohedral.
Example: a rhombic icosahedron is an isozonohedron but not an isohedron.
Examples
Classes of isohedra by symmetry
''k''-isohedral{{anchor|monohedral}} figure
A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domain.
Similarly a k-isohedral tiling has k separate symmetry orbits (and may contain m different shaped faces for some m < k).Craig S. Kaplan.
"Introductory Tiling Theory for Computer Graphics".
2009.
Chapter 5 "Isohedral Tilings".
p.
35.
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).Tilings and Patterns, p.20, 23
Here are some example k-isohedral polyhedra and tilings, with their faces colored by their k symmetry positions:
Related terms
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.http://www.polytope.net/hedrondude/dice4.htm
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.
An isotopic 2-dimensional figure is isotoxal (edge-transitive).
An isotopic 3-dimensional figure is isohedral (face-transitive).
An isotopic 4-dimensional figure is isochoric (cell-transitive).
See also
Edge-transitive
Anisohedral tiling
Notes
References
Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, , p. 367 Transitivity
External links
isohedra 25 classes of isohedra with a finite number of sides
Dice Design at The Dice Lab
