In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (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 mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations,

rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make

fair dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ga ...

. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...

number of faces. The

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of an isohedral polyhedron is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

, 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 are isohedral. For example, a rhombic icosahedron is an isozonohedron but not an isohedron.

Examples

Classes of isohedra by symmetry

''k''-isohedral figure

A polyhedron (or polytope in general) is ''k''-isohedral if it contains ''k'' faces within its symmetry fundamental domains. Similarly, a ''k''-isohedral tiling has ''k'' separate symmetry orbits (it may contain ''m'' different face shapes, for ''m'' = ''k'', or only for some ''m'' < ''k''). ("1-isohedral" is the same as "isohedral".) A monohedral polyhedron or

monohedral tiling A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

(''m'' = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An ''m''-hedral polyhedron or tiling has ''m'' different face shapes ("''dihedral''", "''trihedral''"... are the same as "2-hedral", "3-hedral"... respectively). Here are some examples of ''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'' ≥ 4) or ''n''- honeycomb (''n'' ≥ 3) that has its

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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. A facet-transitive or isotopic figure is an ''n''-dimensional polytope or honeycomb with its facets ((''n''−1)- faces) congruent and transitive. The

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, i.e. edge-transitive. *An isotopic 3-dimensional figure is isohedral, i.e. face-transitive. *An isotopic 4-dimensional figure is isochoric, i.e. cell-transitive.

References

External links