Vertex configurationIn geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.
Isotoxal figureIn geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged. An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal.
Arrangement of linesIn geometry, an arrangement of lines is the subdivision of the plane formed by a collection of lines. Problems of counting the features of arrangements have been studied in discrete geometry, and computational geometers have found algorithms for the efficient construction of arrangements. Intuitively, any finite set of lines in the plane cuts the plane into two-dimensional polygons (cells), one-dimensional line segments or rays, and zero-dimensional crossing points.
Uniform tilingIn geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain.
List of Euclidean uniform tilingsThis table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.
Isohedral figureIn 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.
Trihexagonal tilingIn geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.
Truncated trihexagonal tilingIn geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}. There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares.
Rhombic dodecahedronIn geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly times the short face-diagonal length; thus, the acute angles on each face measure arccos(1/3), or approximately 70.53°.
Quasiregular polyhedronIn geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.
