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7-demicubic Honeycomb
The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb. It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h and the alternated vertices create 7-orthoplex facets. D7 lattice The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice. The 84 vertices of the rectified 7-orthoplex vertex figure of the ''7-demicubic honeycomb'' reflect the kissing number 84 of this lattice. The best known is 126, from the E7 lattice and the 331 honeycomb. The D packing (also called D) can be constructed by the union of two ''D7 lattices''. The D packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 7-demicubic lattices: It is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 7-honeycomb
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope facets around each peak. There are exactly three such convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This ina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 7-orthoplex
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex. There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex. Rectified 7-orthoplex The ''rectified 7-orthoplex'' is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb. : or Alternate names * rectified heptacross * rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon Images Construction There are two Coxeter groups asso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. The simplest case In the simplest case, shown in the first picture, we are given a finite set of points in the Euclidea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tritruncated 7-orthoplex
In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex. There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube. Truncated 7-orthoplex Alternate names * Truncated heptacross * Truncated hecatonicosoctaexon (Jonathan Bowers) Coordinates Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of : (±2,±1,0,0,0,0,0) Images Construction There are two Coxeter groups associated with the ''truncated 7-orthoplex'', one with the C7 or ,35Coxeter group, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadritruncated 7-cubic Honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol 7. Related honeycombs The ,35,4 , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb. The ''7-cubic honeycomb'' can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets. Quadr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Tessellation
Voronoi or Voronoy is a Slavic masculine surname; its feminine counterpart is Voronaya. It may refer to *Georgy Voronoy (1868–1908), Russian and Ukrainian mathematician **Voronoi diagram **Weighted Voronoi diagram ** Voronoi deformation density **Voronoi formula **Voronoi pole In geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram. Definition Let V be the Voronoi diagram for a set of sites P, and let V_p be the Voronoi cell of V corresponding to a site ... ** Centroidal Voronoi tessellation {{Disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-cube Honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol 7. Related honeycombs The ,35,4 , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb. The ''7-cubic honeycomb'' can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets. Quadritr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Body Centered Cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc, and alternatively called ''cubic close-packed'' or ccp) Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive unit cells often are not. Bravais lattices The three Bravais lattices in the cubic crystal system are: The primitive cubic lattice (cP) consists of one lattice point on each corner of the cube; this means each simple cubic unit cell has in total one lattice point. Each atom at a lattice point is then shared equally between eight adjacent cube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3 31 Honeycomb
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. Construction It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the short branch leaves the 6-simplex facet: : Removing the node on the end of the 3-length branch leaves the 321 facet: : The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope. : The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131). : The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031). : The cell figure is determined by removing the ringed node of the face ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E7 Lattice
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. Construction It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the short branch leaves the 6-simplex facet: : Removing the node on the end of the 3-length branch leaves the 321 facet: : The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope. : The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131). : The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031). : The cell figure is determined by removing the ringed node of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
