In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or facetransitive 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 edgetransitive (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 vertextransitive (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 (isogonal, isotoxal, and isohedral). A polyhedron which is isohedral and isogonal is said to be noble.
Not all isozonohedra^{[2]} are isohedral.^{[3]} Example: a rhombic icosahedron is an isozonohedron but not an isohedron.^{[4]}
Convex  Concave  

The hexagonal bipyramid, V4.4.6 is a nonregular example of an isohedral polyhedron. 
The isohedral Cairo pentagonal tiling, V3.3.4.3.4 
The rhombic dodecahedral honeycomb is an example of an isohedral (and isochoric) spacefilling honeycomb. 
Topological square tiling distorted into spiraling H shapes. 
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 domain.^{[5]}
Similarly a kisohedral tiling has k separate symmetry orbits (and may contain m different shaped faces for some m < k).^{[6]}
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 rhedral polyhedra or tiling has r types of faces (also called dihedral, trihedral for 2 or 3 respectively).^{[7]}
Here are some example 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 type of triangle and 2 types of squares  The pseudorhombicuboctahedron has 1 type of triangle and 3 types of squares.  The deltoidal icositetrahedron has with 1 type of face.  The pseudodeltoidal icositetrahedron has 2 types of identicalshaped faces. 
2isohedral  4isohedral  Isohedral  3isohedral 

(2hedral) regularfaced tilings  Monohedral tilings  
The Pythagorean tiling has 2 sizes of squares.  This 3uniform tiling has 3 types identicalshaped triangles and 1 type of square.  The herringbone pattern has 1 type of rectangular face.  This pentagonal tiling has 3 types of identicalshaped irregular pentagon faces. 
A celltransitive or isochoric figure is an npolytope (n > 3) or honeycomb that has its cells congruent and transitive with each other. In 3dimensional honeycombs, the catoptric honeycombs, duals to the uniform honeycombs are isochoric. In 4dimensions, isochoric polytopes have been enumerated up to 20 cells.^{[8]}
A facettransitive or isotopic figure is a ndimensional 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.