≥2-dimensional tessellation or ≥3-dimensional polytope with identical faces
"isohedron" redirects here. Not to be confused with icosahedron.
For the related Isohedral numbers, see Anisohedral tiling.
A set of isohedral dice
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 face-transitive 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 vertex-transitive, 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 self-dual or dual with another Platonic solid, are vertex-, edge-, and face-transitive (i.e. isogonal, isotoxal, and isohedral).
A form that is isohedral, has regular vertices, and is also edge-transitive (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]
^McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR 3619822, S2CID 195047512.
^Weisstein, Eric W. "Isozonohedron". mathworld.wolfram.com. Retrieved 2019-12-26.
^Weisstein, Eric W. "Isohedron". mathworld.wolfram.com. Retrieved 2019-12-21.
^Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-21.
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...
They are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, and is 2-isohedral keeping chiral...
ratio) as its faces. The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces. The rhombic hexecontahedron...
order 48. All faces are squares. Trigonal trapezohedron (also called isohedral rhombohedron): with D3d symmetry, order 12. All non-obtuse internal angles...
and isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, are isohedral and isotoxal, but not...
called the great disnub dirhombidodecacron. It is a nonconvex infinite isohedral polyhedron. Like the visually identical great dirhombicosidodecacron in...
faces of the three octahedra; it is another of the known isohedral deltahedra. A third isohedral deltahedron sharing the same face planes, the compound...
two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent [...]), and vice...
rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually...
topologically equivalent variation, sometimes called a deltoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites...
trapezohedron, which has six congruent rhombic faces (also called an isohedral rhombohedron). For parallelepipeds with D2h symmetry, there are two cases:...
or of three octagonal prisms. The great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21)...
faces in common). The great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron...
symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be...
internal angle of a regular pentagon, 3π/5, is not a divisor of 2π. An isohedral tiling is a special variation of a monohedral tiling in which all tiles...
(having the hexagonal faces in common). The rhombicosacron is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has...
icosidodecahedron is the great rhombic triacontahedron; it is nonconvex, isohedral and isotoxal. It has 30 intersecting rhombic faces. It can also be called...
known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal. k-uniform tilings with...
Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract...
Critchlow, p.74-75, pattern 1 Tilings and Patterns, from list of 107 isohedral tilings, p.473-481 Coxeter, Regular Complex Polytopes, pp. 111-112, p...
disnub dirhombidodecahedron (Skilling's figure). Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and...
monohedral tilings it is denoted [3.6.3.6]. It is also one of 56 possible isohedral tilings by quadrilaterals, and one of only eight tilings of the plane...
={\frac {\pi ^{\frac {1}{3}}(6V)^{\frac {2}{3}}}{A}}\approx 0.98163} Isohedral variations can be constructed with pentagonal faces with 3 edge lengths...
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