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Bitruncated Tesseractic Honeycomb
In four-dimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a bitruncation of a tesseractic honeycomb. It is also called a cantic quarter tesseractic honeycomb from its q2 construction. Other names * Bitruncated tesseractic tetracomb (batitit) Related honeycombs See also Regular and uniform honeycombs in 4-space: * Tesseractic honeycomb * Demitesseractic honeycomb * 24-cell honeycomb * Truncated 24-cell honeycomb * Snub 24-cell honeycomb * 5-cell honeycomb * Truncated 5-cell honeycomb * Omnitruncated 5-cell honeycomb Notes References * Kaleidoscopes: Selected Writings of H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ..., edite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of Coxeter groups, symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams. History of discovery *Regular polytopes: (convex faces) **1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. *Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) **1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
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 spaces, higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and ''semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An ''aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern. A ''tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncated 5-cell Honeycomb
In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1. Structure Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell. Alternate names * Cyclopentachoric tetracomb * Pentachoric-dispentachoric tetracomb Projection by folding The ''5-cell honeycomb'' can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement: A4 lattice The vertex arrangemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 5-cell Honeycomb
In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1. Structure Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell. Alternate names * Cyclopentachoric tetracomb * Pentachoric-dispentachoric tetracomb Projection by folding The ''5-cell honeycomb'' can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement: A4 lattice The vertex arrangement of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cell Honeycomb
In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1. Structure Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell. Alternate names * Cyclopentachoric tetracomb * Pentachoric-dispentachoric tetracomb Projection by folding The ''5-cell honeycomb'' can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement: A4 lattice The vertex arrangement of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub 24-cell Honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra. It can be seen as an alternation of a truncated 24-cell honeycomb, and can be represented by Schläfli symbol s, s, and 3 other snub constructions. It is defined by an irregular decachoron vertex figure (10-celled 4-polytope), faceted by four snub 24-cells, one 16-cell, and five 5-cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at mid-edges into a central octahedron and four corner tetrahedra. Then the four side-facets of the prism, the triangular prisms become tridiminished icosahedra. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 24-cell Honeycomb
In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells. It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the _4 construction. This truncated 24-cell has Schläfli symbol t, and its snub is represented as s. Alternate names * Truncated icositetrachoric tetracomb * Truncated icositetrachoric honeycomb * Cantitruncated 16-cell honeycomb * Bicantitruncated tesseractic honeycomb Symmetry constructions There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators. See also Regular and uniform honeycombs in 4-space: *Tesseractic honeycomb *16 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
24-cell Honeycomb
In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol . The dual tessellation by regular 16-cell honeycomb has Schläfli symbol . Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space. Coordinates The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D4 or F4 root lattice. Each 24-cell is then centered at a D4 lattice point, i.e. one of :\left\. These points can also be described as Hurwitz quaternions with even square norm. The vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with odd square norm. It can be constructed as a birectified tesseractic honeycomb, by taking a tesseractic honeycomb and placing vertices at the centers of all the squa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demitesseractic Honeycomb
In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of 16-cell facets, three around every face. Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice. Alternate names * Hexadecachoric tetracomb/honeycomb * Demitesseractic tetracomb/honeycomb Coordinates Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even. D4 lattice The vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice. The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003. The related D lattice (also called D) can be construc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tesseractic Honeycomb
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of tesseract facets. Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex. It is an analog of the square tiling, , of the plane and the cubic honeycomb, , of 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form . Tessellations in this family are Self-dual. Coordinates Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l). Sphere packing Like all regular hypercubic honeycombs, the tesseractic honeycomb corresponds to a sphere packing of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic honeycomb of 4 dimensions, verte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
