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Skew Polytope
In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces. Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra. History According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to ''regular skew polyhedra''. Coxeter offered a modified Schläfli symbol for these figures, with implying the vertex figure, -gons around a vertex, and -gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes. The regular skew polyhedra, represented by , follow this equation: : 2 \cos \frac \cos \frac = \cos \frac A first set , repeats the five convex Platonic solids, and one nonconvex Kepler–Poinsot solid: : Fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron. The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with 1, unit s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Polyhedron Almost Regular 42 Vertices
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitruncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two degrees of truncations, including a bitruncation. Truncated 24-cell The truncated 24-cell or truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure. Construction The truncated 24-cell can be constructed from polytopes with three symmetry groups: *F4 ,4,3 A truncation of the 24-cell. *B4 ,3,4 A cantitruncation of the 16-cell, with two families of truncated octahedral cells. *D4 1,1,1 An omnitruncation of the demitesseract, with three families of truncated octahedral cells. Zonotope It is also a zonotope: it can be formed as the Minkowski sum of the six line segments connecting opposite pairs among the twelve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 24-cell
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell. There are 3 unique degrees of runcinations of the 24-cell including with permutations truncations and cantellations. Runcinated 24-cell In geometry, the Runcination (geometry), runcinated 24-cell or small prismatotetracontoctachoron is a uniform 4-polytope bounded by 48 octahedron, octahedra and 192 triangular prisms. The octahedral cells correspond with the cells of a 24-cell and its dual. Emanuel Lodewijk Elte, E. L. Elte identified it in 1912 as a semiregular polytope. Alternate names * Runcinated 24-cell (Norman Johnson (mathematician), Norman W. Johnson) * Runcinated icositetrachoron * Runcinated polyoctahedron * Small prismatotetracontoctachoron (spic) (Jonathan Bowers) Coordinates The Cartesian coordinates of the runcinated 24-cell having edge length 2 is given by all permutations of sign and coordinates of: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitruncated 5-cell
In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron. Construction The truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices. Structure The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces. Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 5-cell
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to Face (geometry), face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including with permutations, truncations, and cantellations. Runcinated 5-cell The runcinated 5-cell or small prismatodecachoron is constructed by Expansion (geometry), expanding the cell (mathematics), cells of a 5-cell radially and filling in the gaps with triangular prism (geometry), prisms (which are the face prisms and edge figures) and tetrahedron, tetrahedra (cells of the dual 5-cell). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual. Topologically, under its highest symmetry, 3,3,3, there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms. The rectangles are always squares because the two pairs of edges correspond to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-simplex T12
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, hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Plane
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number of an irreducible root system. *The Coxeter number is the order of any Coxeter element;. *The Coxeter number is where is the rank, and is the number of reflections. In the crystallographic case, is half the number of roots; and is the dimension of the corresponding semisimple Lie algebra. *If the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
