geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ge ...

of dimension (a plane tiling) or higher, or a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

of dimension (a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

) 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 Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...

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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

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 fa ...

, i.e. isogonal. The

Catalan solids Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #1 ...

, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The

Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

, 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 In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this mea ...

(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 (''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''-

(''n'' ≥ 4) or ''n''-

honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey t ...

(''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. 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