Search results
Results from the WOW.Com Content Network
A cell-transitive or isochoric figure is an n-polytope (n ≥ 4) or n-honeycomb (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.
If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. [19]
It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs.
Facet, an (n-1)-dimensional element; Ridge, an (n-2)-dimensional element; Peak, an (n-3)-dimensional element; For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure: not itself an element of a polytope, but a diagram showing how the elements meet.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}.
An n-apeirotope is an infinite n-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc. There are two main geometric classes of apeirotope: [15] Regular honeycombs in n dimensions, which completely fill an n-dimensional space.
Dual semi-regular Article Face configuration Schläfli symbol Image Apeirogonal deltohedron: V3 3.∞ : dsr{2,∞} Apeirogonal bipyramid: V4 2.∞ : dt{2,∞} Cairo pentagonal tiling
F, L, N, P, and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image. Their symmetry group consists only of the identity mapping. T, and U can be oriented in 4 ways by rotation. They have an axis of reflection aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line ...