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D5 Lattice
The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb. It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h and the alternated vertices create 5-orthoplex facets. D5 lattice The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the ''5-demicubic honeycomb'' reflect the kissing number 40 of this lattice. The D packing (also called D) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). : � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 5-honeycomb
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex , the 6-cube (hexeract) , and the 6-orthoplex (hexacross) . 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) ** 190 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of ''semiregular'' polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures ( star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well. Operations Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombic ... [...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] |
Bitruncated 5-orthoplex
In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube. Truncated 5-orthoplex Alternate names * Truncated pentacross * Truncated triacontiditeron (Acronym: tot) (Jonathan Bowers) Coordinates Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of : (±2,±1,0,0,0) Images The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge. Bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tritruncated 5-cubic Honeycomb
In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic honeycomb''. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space. Constructions There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol 5. Related polytopes and honeycombs The ,33,4 , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic hone ... [...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] |
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] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Camb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kissing Number
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number. In general, the kissing number problem seeks the maximum possible kissing number for ''n''-dimensional spheres in (''n'' + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space. Finding the kissing number when centers of spheres are confined to a line (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 5-orthoplex
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex. There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex. Rectified 5-orthoplex Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups. E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5-d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the ''packing density'' of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, the densest packing uses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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