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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.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
Let be a metric space with distance function .Let be a set of indices and let () be a tuple (indexed collection) of nonempty subsets (the sites) in the space .The Voronoi cell, or Voronoi region, , associated with the site is the set of all points in whose distance to is not greater than their distance to the other sites , where is any index different from .
The fundamental region is a shape such as a rectangle that is repeated to form the tessellation. [22] For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex. [18] The sides of the polygons are not necessarily identical to the edges of the tiles.
Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools.
Example tessellation based on a Type 1 hexagonal tile. In its simplest form, the criterion simply states that any hexagon with a pair of opposite sides that are parallel and congruent will tessellate the plane. [8] In Gardner's article, this is called a type 1 hexagon. [7] This is also true of parallelograms.
Polyhedra are the three dimensional correlates of polygons. They are built from flat faces and straight edges and have sharp corner turns at the vertices. Although a cube is the only regular polyhedron that admits of tessellation, many non-regular 3-dimensional shapes can tessellate, such as the truncated octahedron.
A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3-space, [1] also called the Archimedean honeycombs.