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Demienneract
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM9 for a 9-dimensional ''half measure'' polytope. Coxeter named this polytope as 161 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract: : (±1,±1,±1,±1,±1,±1,±1,±1,±1) with an odd number of plus signs. Images References * H.S.M. Coxeter: ** Coxeter, ''Regular Polytopes'', (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p.& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demienneract Ortho Petrie
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM9 for a 9-dimensional ''half measure'' polytope. Coxeter named this polytope as 161 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract: : (±1,±1,±1,±1,±1,±1,±1,±1,±1) with an odd number of plus signs. Images References * H.S.M. Coxeter: ** Coxeter, ''Regular Polytopes'', (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p.& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alternate names It can also be called a heptapeton, or hepta-6-tope, as a 7- facetted polytope in 6-dimensions. The name ''heptapeton'' is derived from ''hepta'' for seven facets in Greek and ''-peta'' for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym hop. As a configuration This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin7 & 6 & 15 & 20 & 15 & 6 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-simplex T0
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertex corners. The tetrahedron is the simplest of all the ordinary convex polytope, convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean geometry, Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid (geometry), pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...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] |
5-cell
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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Graph 4
A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a saltire in heraldic terminology. The cross has been widely recognized as a symbol of Christianity from an early period.''Christianity: an introduction'' by Alister E. McGrath 2006 pages 321-323 However, the use of the cross as a religious symbol predates Christianity; in the ancient times it was a pagan religious symbol throughout Europe and western Asia. The effigy of a man hanging on a cross was set up in the fields to protect the crops. It often appeared in conjunction with the female-genital circle or oval, to signify the sacred marriage, as in Egyptian amule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid .Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68 It is a part of an infinite family of polytopes, called cross-polytopes or ''orthoplexes'', and is analogous to the octahedron in three dimensions. It is Coxeter's \beta_4 polytope. Conway's name for a cross-polytope is orthoplex, for ''orthant complex''. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices. Geometry The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). Each of its 4 successor convex regular 4-polytopes can be constructed as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-simplex T0
In five-dimensional geometry, a 5- simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5-simplex is a solution to the problem: ''Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.'' Alternate names It can also be called a hexateron, or hexa-5-tope, as a 6- facetted polytope in 5-dimensions. The name ''hexateron'' is derived from ''hexa-'' for having six facets and '' teron'' (with ''ter-'' being a corruption of ''tetra-'') for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix. As a configuration This configuration matrix represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-simplex. The nondiagonal num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5-simplex is a solution to the problem: ''Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.'' Alternate names It can also be called a hexateron, or hexa-5-tope, as a 6- facetted polytope in 5-dimensions. The name ''hexateron'' is derived from ''hexa-'' for having six facets and '' teron'' (with ''ter-'' being a corruption of ''tetra-'') for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix. As a configuration This configuration matrix represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-simplex. The nondiagonal numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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