In geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or facetransitive if 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 two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or 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. An isohedron has an even number of faces.
The dual of an isohedral polyhedron is vertextransitive, i.e. 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 selfdual or dual with another Platonic solid, are vertex, edge, and facetransitive (i.e. isogonal, isotoxal, and isohedral).
A form that is isohedral, has regular vertices, and is also edgetransitive (i.e. isotoxal) 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.
A polyhedron which is isohedral and isogonal is said to be noble.
Not all isozonohedra^{[2]} are isohedral.^{[3]} For example, a rhombic icosahedron is an isozonohedron but not an isohedron.^{[4]}
Convex  Concave  

Hexagonal bipyramids, V4.4.6, are nonregular isohedral polyhedra. 
The Cairo pentagonal tiling, V3.3.4.3.4, is isohedral. 
The rhombic dodecahedral honeycomb is isohedral (and isochoric, and spacefilling). 
A square tiling distorted into a spiraling H tiling (topologically equivalent) is still isohedral. 
Faces  Face config. 
Class  Name  Symmetry  Order  Convex  Coplanar  Nonconvex 

4  V3^{3}  Platonic  tetrahedron tetragonal disphenoid rhombic disphenoid 
T_{d}, [3,3], (*332) D_{2d}, [2^{+},2], (2*) D_{2}, [2,2]^{+}, (222) 
24 4 4 4 

6  V3^{4}  Platonic  cube trigonal trapezohedron asymmetric trigonal trapezohedron 
O_{h}, [4,3], (*432) D_{3d}, [2^{+},6] (2*3) D_{3} [2,3]^{+}, (223) 
48 12 12 6 

8  V4^{3}  Platonic  octahedron square bipyramid rhombic bipyramid square scalenohedron 
O_{h}, [4,3], (*432) D_{4h},[2,4],(*224) D_{2h},[2,2],(*222) D_{2d},[2^{+},4],(2*2) 
48 16 8 8 

12  V3^{5}  Platonic  regular dodecahedron pyritohedron tetartoid 
I_{h}, [5,3], (*532) T_{h}, [3^{+},4], (3*2) T, [3,3]^{+}, (*332) 
120 24 12 

20  V5^{3}  Platonic  regular icosahedron  I_{h}, [5,3], (*532)  120  
12  V3.6^{2}  Catalan  triakis tetrahedron  T_{d}, [3,3], (*332)  24  
12  V(3.4)^{2}  Catalan  rhombic dodecahedron deltoidal dodecahedron 
O_{h}, [4,3], (*432) T_{d}, [3,3], (*332) 
48 24 

24  V3.8^{2}  Catalan  triakis octahedron  O_{h}, [4,3], (*432)  48  
24  V4.6^{2}  Catalan  tetrakis hexahedron  O_{h}, [4,3], (*432)  48  
24  V3.4^{3}  Catalan  deltoidal icositetrahedron  O_{h}, [4,3], (*432)  48  
48  V4.6.8  Catalan  disdyakis dodecahedron  O_{h}, [4,3], (*432)  48  
24  V3^{4}.4  Catalan  pentagonal icositetrahedron  O, [4,3]^{+}, (432)  24  
30  V(3.5)^{2}  Catalan  rhombic triacontahedron  I_{h}, [5,3], (*532)  120  
60  V3.10^{2}  Catalan  triakis icosahedron  I_{h}, [5,3], (*532)  120  
60  V5.6^{2}  Catalan  pentakis dodecahedron  I_{h}, [5,3], (*532)  120  
60  V3.4.5.4  Catalan  deltoidal hexecontahedron  I_{h}, [5,3], (*532)  120  
120  V4.6.10  Catalan  disdyakis triacontahedron  I_{h}, [5,3], (*532)  120  
60  V3^{4}.5  Catalan  pentagonal hexecontahedron  I, [5,3]^{+}, (532)  60  
2n  V3^{3}.n  Polar  trapezohedron asymmetric trapezohedron 
D_{nd}, [2^{+},2n], (2*n) D_{n}, [2,n]^{+}, (22n) 
4n 2n 


2n 4n 
V4^{2}.n V4^{2}.2n V4^{2}.2n 
Polar  regular nbipyramid isotoxal 2nbipyramid 2nscalenohedron 
D_{nh}, [2,n], (*22n) D_{nh}, [2,n], (*22n) D_{nd}, [2^{+},2n], (2*n) 
4n 
A polyhedron (or polytope in general) is kisohedral if it contains k faces within its symmetry fundamental domains.^{[5]} Similarly, a kisohedral tiling has k separate symmetry orbits (it may contain m different face shapes, for m = k, or only for some m < k).^{[6]} ("1isohedral" is the same as "isohedral".)
A monohedral polyhedron or monohedral tiling (m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An mhedral polyhedron or tiling has m different face shapes ("dihedral", "trihedral"... are the same as "2hedral", "3hedral"... respectively).^{[7]}
Here are some examples of kisohedral polyhedra and tilings, with their faces colored by their k symmetry positions:
3isohedral  4isohedral  isohedral  2isohedral 

2hedral regularfaced polyhedra  Monohedral polyhedra  
The rhombicuboctahedron has 1 triangle type and 2 square types.  The pseudorhombicuboctahedron has 1 triangle type and 3 square types.  The deltoidal icositetrahedron has 1 face type.  The pseudodeltoidal icositetrahedron has 2 face types, with same shape. 
2isohedral  4isohedral  Isohedral  3isohedral 

2hedral regularfaced tilings  Monohedral tilings  
The Pythagorean tiling has 2 square types (sizes).  This 3uniform tiling has 3 triangle types, with same shape, and 1 square type.  The herringbone pattern has 1 rectangle type.  This pentagonal tiling has 3 irregular pentagon types, with same shape. 
A celltransitive or isochoric figure is an npolytope (n ≥ 4) or nhoneycomb (n ≥ 3) that has its cells congruent and transitive with each others. In 3 dimensions, the catoptric honeycombs, duals to the uniform honeycombs, are isochoric. In 4 dimensions, isochoric polytopes have been enumerated up to 20 cells.^{[8]}
A facettransitive or isotopic figure is an ndimensional polytope 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.