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5 21 Honeycomb
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.Coxeter, 1973, Chapter 5: The Kaleidoscope By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by Maryna Viazovska in 2016 using the theory of modular forms. Viazovska was awarded the Fields Medal for this work in 2022. This honeycomb was first studied by Gosset who called it a ''9-ic semi-regular figure'' (Gosset regarded honeycombs in ''n'' dimensions as degenerate ''n''+1 polytopes). Each vertex of the 521 honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplicies. The vertex figure of Gosset's honeycomb is the semiregular 421 polytope. It is the final figure in the k21 family. This honeycomb is highly regular in the sense that its symmetry group (the affine _8 Weyl group) acts transitively on the ''k''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9-polytope
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets. A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets. Regular 9-polytopes Regular 9-polytopes can be represented by the Schläfli symbol , with w 8-polytope facets around each peak. There are exactly three such convex regular 9-polytopes: # - 9-simplex # - 9-cube # - 9-orthoplex There are no nonconvex regular 9-polytopes. Euler characteristic The topology of any given 9-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, whatever their underlying topology. This inadequacy of the Euler characteristic to rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-simplex T0
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's \alpha_4 polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides. The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosset 4 21 Polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an ''8-ic semi-regular figure''.Gosset, 1900 Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, . The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421. These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: . 421 polytope The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 321 po ... [...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] |
E7 Graph
E7, E07, E-7 or E7 may refer to: Science and engineering * E7 liquid crystal mixture * E7 (mathematics), E7, the Lie group in mathematics * E7 polytope, E7 polytope, in geometry * E7 papillomavirus protein * E7 European long distance path Transport * EMD E7, a diesel locomotive * European route E07, an international road * Peugeot E7, a hackney cab * PRR E7, a steam locomotive * Carbon Motors E7,a police car * E7 series, a Japanese high-speed train * Nihonkai-Tōhoku Expressway and Akita Expressway (between Kawabe JCT and Kosaka JCT), route E7 in Japan * Cheras–Kajang Expressway, route E7 in Malaysia Other uses * ''Boeing E-7'', either: ** Boeing E-7 ARIA, the original designation assigned by the United States Air Force under the Mission Designation System to the EC-18B Advanced Range Instrumentation Aircraft. ** Boeing E-7 Wedgetail, the designation assigned by the Royal Australian Air Force to the Boeing 737 AEW&C (airborne early warning and control) aircraft. * Economy 7, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosset 3 21 Polytope
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.Gosset, 1900 Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences. The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is constructed by points at the tetrahedral centers of the 321, and is the same as the rectified 132. These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . 321 polytope In 7-dimensional geometry, the 321 is a uniform polytope. It has 56 vertices, and 702 facets: 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
