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5-simplex T01 A3
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 numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Five-dimensional Space
A five-dimensional space is a space with five dimensions. In mathematics, a sequence of ''N'' numbers can represent a location in an ''N''-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. Whether or not the universe is five-dimensional is a topic of debate. Physics Much of the early work on five-dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity and electromagnetism. German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century. To explain why this dimension would no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tritruncated 6-simplex
In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex. Truncated 6-simplex Alternate names * Truncated heptapeton (Acronym: til) (Jonathan Bowers) Coordinates The vertices of the ''truncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex. Images Bitruncated 6-simplex Alternate names * Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers) Coordinates The vertices of the ''bitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitruncated 6-cube
In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube. Truncated 6-cube Alternate names * Truncated hexeract (Acronym: tox) (Jonathan Bowers) Construction and coordinates The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at 1/(\sqrt+2) of the edge length. A regular 5-simplex replaces each original vertex. The Cartesian coordinates of the vertices of a ''truncated 6-cube'' having edge length 2 are the permutations of: :\left(\pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right) Images Related polytopes The '' truncated 6- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitruncation
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation or In regular polyhedra and tilings For regular polyhedra (i.e. regular 3-polytopes), a ''bitruncated'' form is the truncated dual. For example, a bitruncated cube is a truncated octahedron. In regular 4-polytopes and honeycombs For a regular 4-polytope, a ''bitruncated'' form is a dual-symmetric operator. A bitruncated 4-polytope is the same as the bitruncated dual, and will have double the symmetry if the original 4-polytope is self-dual. A regular polytope (or honeycomb) will have its cells bitruncated into truncated cells, and the vertices are replaced by truncated cells. Self-dual 4-polytope/honeycombs An interesting result of this operation is that self- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 6-cube
In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube. Truncated 6-cube Alternate names * Truncated hexeract (Acronym: tox) (Jonathan Bowers) Construction and coordinates The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at 1/(\sqrt+2) of the edge length. A regular 5-simplex replaces each original vertex. The Cartesian coordinates of the vertices of a ''truncated 6-cube'' having edge length 2 are the permutations of: :\left(\pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right) Images Related polytopes The '' truncated 6- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-polytope
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. Definition A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope. Furthermore, the following requirements must be met: * Each 4-face must join exactly two 5-faces (facets). * Adjacent facets are not in the same five-dimensional hyperplane. * The figure is not a compound of other figures which meet the requirements. Characteristics The topology of any given 6-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 us ... [...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] |
Apex (geometry)
In geometry, an apex (plural apices) is the vertex which is in some sense the "highest" of the figure to which it belongs. The term is typically used to refer to the vertex opposite from some " base". The word is derived from the Latin for 'summit, peak, tip, top, extreme end'. Isosceles triangles In an isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ..., the apex is the vertex where the two sides of equal length meet, opposite the unequal third side. Pyramids and cones In a Pyramid (geometry), pyramid or Cone (geometry), cone, the apex is the vertex at the "top" (opposite the base). In a pyramid, the vertex is the point that is part of all the lateral faces, or where all the lateral edges meet. References {{elementary-geometry-stub Parts of a triangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlegel Diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes. Construction The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows: :A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic Projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to the diameter through the point. It is a smooth function, smooth, bijection, bijective function (mathematics), function from the entire sphere except the center of projection to the entire plane. It maps circle of a sphere, circles on the sphere to generalised circle, circles or lines on the plane, and is conformal map, conformal, meaning that it preserves angles at which curves meet and thus Local property, locally approximately preserves similarity (geometry), shapes. It is neither isometry, isometric (distance preserving) nor Equiareal map, equiareal (area preserving). The stereographic projection gives a way to representation (mathematics), represent a sphere by a plane. The metric tensor, metric induced metric, induced by the inverse s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
