1.
2 21 polytope
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In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 6-ic semi-regular figure. It is also called the Schläfli polytope and its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, the rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the face centers of the 221. The 221 has 27 vertices, and 99 facets,27 5-orthoplexes and 72 5-simplices and its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon and its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements can also be extracted and drawn on this projection, the Schläfli graph contains the 1-skeleton of this polytope. E. L. Elte named it V27 in his 1912 listing of semiregular polytopes, icosihepta-heptacontidi-peton - 27-72 facetted polypeton The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope, Its construction is based on the E6 group. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the 5-simplex. Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form, every simplex facet touches an 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node, vertices are colored by their multiplicity in this projection, in progressive order, red, orange, yellow. The number of vertices by color are given in parentheses, the 221 is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. This can be seen in the Coxeter plane projections, the 24 vertices of the 24-cell are projected in the same two rings as seen in the 221. This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram, the regular complex polygon 333, in C2 has a real representation as the 221 polytope, in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess and it has 27 vertices,72 3-edges, and 2733 faces. Its complex reflection group is 333, order 648, the 221 is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes
2.
Rectified 2 21 polytope
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In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 6-ic semi-regular figure. It is also called the Schläfli polytope and its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, the rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the face centers of the 221. The 221 has 27 vertices, and 99 facets,27 5-orthoplexes and 72 5-simplices and its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon and its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements can also be extracted and drawn on this projection, the Schläfli graph contains the 1-skeleton of this polytope. E. L. Elte named it V27 in his 1912 listing of semiregular polytopes, icosihepta-heptacontidi-peton - 27-72 facetted polypeton The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope, Its construction is based on the E6 group. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the 5-simplex. Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form, every simplex facet touches an 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node, vertices are colored by their multiplicity in this projection, in progressive order, red, orange, yellow. The number of vertices by color are given in parentheses, the 221 is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. This can be seen in the Coxeter plane projections, the 24 vertices of the 24-cell are projected in the same two rings as seen in the 221. This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram, the regular complex polygon 333, in C2 has a real representation as the 221 polytope, in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess and it has 27 vertices,72 3-edges, and 2733 faces. Its complex reflection group is 333, order 648, the 221 is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes
3.
Projection (linear algebra)
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. That is, whenever P is applied twice to any value, though abstract, this definition of projection formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on an object by examining the effect of the projection on points in the object. For example, the function maps the point in three-dimensional space R3 to the point is an orthogonal projection onto the x–y plane. This function is represented by the matrix P =, the action of this matrix on an arbitrary vector is P =. To see that P is indeed a projection, i. e. P = P2, a simple example of a non-orthogonal projection is P =. Via matrix multiplication, one sees that P2 = = = P. proving that P is indeed a projection, the projection P is orthogonal if and only if α =0. Let W be a finite dimensional space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively, then P has the following properties, By definition, P is idempotent. P is the identity operator I on U ∀ x ∈ U, P x = x and we have a direct sum W = U ⊕ V. Every vector x ∈ W may be decomposed uniquely as x = u + v with u = P x and v = x − P x = x, the range and kernel of a projection are complementary, as are P and Q = I − P. The operator Q is also a projection and the range and kernel of P become the kernel and range of Q and we say P is a projection along V onto U and Q is a projection along U onto V. In infinite dimensional spaces, the spectrum of a projection is contained in as −1 =1 λ I +1 λ P. Only 0 or 1 can be an eigenvalue of a projection, the corresponding eigenspaces are the kernel and range of the projection. Decomposition of a space into direct sums is not unique in general. Therefore, given a subspace V, there may be many projections whose range is V, if a projection is nontrivial it has minimal polynomial x 2 − x = x, which factors into distinct roots, and thus P is diagonalizable. The product of projections is not, in general, a projection, if projections commute, then their product is a projection. When the vector space W has a product and is complete the concept of orthogonality can be used
4.
Coxeter element
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element 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. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections
5.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
6.
Uniform polytope
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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 and this is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures are allowed, which expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs of Euclidean, nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the antiprism in four dimensions. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension and this approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation. Regular n-polytopes have n orders of rectification, the zeroth rectification is the original form. The th rectification is the dual, an extended Schläfli symbol can be used for representing rectified forms, with a single subscript, k-th rectification = tk = kr. Truncation operations that can be applied to regular n-polytopes in any combination, the resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, each higher operation also cuts lower ones too, so a cantellation also truncates vertices. T0,1 or t, Truncation - applied to polygons, a truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges and it can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets, cells are replaced by topologically expanded copies of themselves. There are higher cantellations also, bicantellation t1,3 or r2r, tricantellation t2,4 or r3r, quadricantellation t3,5 or r4r, etc. t0,1,2 or tr, Cantitruncation - applied to polyhedra and higher. It can be seen as a truncation of its rectification, a cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves, runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves, There are higher runcinations also, biruncination t1,4, triruncination t2,5, etc. t0,4 or 2r2r, Sterication - applied to Uniform 5-polytopes and higher. It can be seen as birectifying its birectification, Sterication truncates vertices, edges, faces, and cells, replacing each with new facets
7.
E6 (mathematics)
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The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras. This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, the E6 algebra is thus one of the five exceptional cases. The fundamental group of the form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z. Its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface, the dual representation, which is inequivalent, is also 27-dimensional. In particle physics, E6 plays a role in some grand unified theories, there is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156, the split form, EI, which has maximal compact subgroup Sp/, fundamental group of order 2 and outer automorphism group of order 2. The quasi-split form EII, which has maximal compact subgroup SU × SU/, fundamental group cyclic of order 6, EIII, which has maximal compact subgroup SO × Spin/, fundamental group Z and trivial outer automorphism group. EIV, which has maximal compact subgroup F4, trivial fundamental group cyclic, the EIV form of E6 is the group of collineations of the octonionic projective plane OP2. It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra, the exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. Over finite fields, the Lang–Steinberg theorem implies that H1 =0, meaning that E6 has exactly one twisted form, known as 2E6, the Dynkin diagram for E6 is given by, which may also be drawn as or. Although they span a space, it is much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space. Two 16-dimensional subalgebras that transform as a Weyl spinor of spin and these have a non-zero last entry. 1 generator which is their chirality generator, and is the sixth Cartan generator, the Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU × SU × SU subalgebra. Other maximal subalgebras which have an importance in physics and can be read off the Dynkin diagram, are the algebras of SO × U, in addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional vector representations. The characters of finite dimensional representations of the real and complex Lie algebras, the fundamental representations have dimensions 27,351,2925,351,27 and 78. The E6 polytope is the hull of the roots of E6. It therefore exists in 6 dimensions, its symmetry group contains the Coxeter group for E6 as an index 2 subgroup, the groups of type E6 over arbitrary fields were introduced by Dickson. The points over a field with q elements of the algebraic group E6, whether of the adjoint or simply connected form
8.
Uniform 6-polytope
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In six-dimensional geometry, a uniform polypeton 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 polypeta 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, Regular polytopes,1852, Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes,1940, The search was expanded systematically by H. S. M, Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular uniform star polytopes, Ongoing, Thousands of nonconvex uniform polypeta are known, participating researchers include Jonathan Bowers, Richard Klitzing and Norman Johnson. Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes. Uniform prism There are 6 categorical uniform prisms based on the uniform 5-polytopes, Uniform duoprism There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope, in addition, there are 105 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes. In addition, there are many uniform 6-polytope based on. There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram and they are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex. Bowers-style acronym names are given in parentheses for cross-referencing, the A6 family has symmetry of order 5040. The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, see also list of A6 polytopes for graphs of these polytopes. There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, the B6 family has symmetry of order 46080. They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube, Bowers names and acronym names are given for cross-referencing. See also list of B6 polytopes for graphs of these polytopes, the D6 family has symmetry of order 23040. This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D6 Coxeter-Dynkin diagram, of these,31 are repeated from the B6 family and 16 are unique to this family. The 16 unique forms are enumerated below, bowers-style acronym names are given for cross-referencing
9.
Uniform 5-polytope
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In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets, the complete set of convex uniform 5-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 diagrams, Regular polytopes,1852, Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes, 1940-1988, The search was expanded systematically by H. S. M, Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III. 1966, Norman W. Johnson completed his Ph. D, There are exactly three such regular polytopes, all convex, - 5-simplex - 5-cube - 5-orthoplex There are no nonconvex regular polytopes in 5 or more dimensions. There are 104 known convex uniform 5-polytopes, plus a number of families of duoprism prisms. All except the grand antiprism prism are based on Wythoff constructions, the 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the forms in the B5 family. The bifurcating graph of the D6 family contains the pentacross, as well as a 5-demicube which is an alternated 5-cube, one non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms. That brings the tally to, 19+31+8+46+1=105 In addition there are, Infinitely many uniform 5-polytope constructions based on duoprism prismatic families, Infinitely many uniform 5-polytope constructions based on duoprismatic families, ×, ×, ×. There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings and they are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex. The A5 family has symmetry of order 720,7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440. The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, the B5 family has symmetry of order 3840. This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram, for simplicity it is divided into two subgroups, each with 12 forms, and 7 middle forms which equally belong in both. The 5-cube family of 5-polytopes are given by the hulls of the base points listed in the following table, with all permutations of coordinates. Each base point generates a distinct uniform 5-polytope, all coordinates correspond with uniform 5-polytopes of edge length 2. The D5 family has symmetry of order 1920 and this family has 23 Wythoffian uniform polyhedra, from 3x8-1 permutations of the D5 Coxeter diagram with one or more rings. 15 are repeated from the B5 family and 8 are unique to this family, There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes, This prismatic family has 9 forms, The A1 x A4 family has symmetry of order 240
10.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter 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 tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face
11.
Uniform 1 k2 polytope
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In geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram and it can be named by an extended Schläfli symbol. The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube in 5-dimensions, each polytope is constructed from 1k-1,2 and -demicube facets. Each has a figure of a polytope is a birectified n-simplex. The sequence ends with k=6, as a tessellation of 9-dimensional hyperbolic space. Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings, alicia Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Vol.11, No. 1, pp. 1–24 plus 3 plates,1910, Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings. Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam, H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin,1940 N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Coxeter, Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin,1988 PolyGloss v0.05, Gosset figures
12.
5-demicube
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In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices truncated. It was discovered by Thorold Gosset, since it was the only semiregular 5-polytope, he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2,1 and 1 with a ringed node on one of the short branches, and Schläfli symbol or. It exists in the k21 polytope family as 121 with the Gosset polytopes,221,321, the graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead. Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract and it is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family. There are 23 Uniform 5-polytopes that can be constructed from the D5 symmetry of the demipenteract,8 of which are unique to this family, the 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. In Coxeters notation the 5-demicube is given the symbol 121, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 5D uniform polytopes x3o3o *b3o3o - hin, archived from the original on 4 February 2007
13.
16-cell
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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century and it is also called C16, hexadecachoron, or hexdecahedroid. It is a part of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the tesseract, conways name for a cross-polytope is orthoplex, for orthant complex. The 16-cell has 16 cells as the tesseract has 16 vertices and it is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces,24 edges, and 8 vertices, the 24 edges bound 6 squares lying in the 6 coordinate planes. The eight vertices of the 16-cell are, all vertices are connected by edges except opposite pairs. The Schläfli symbol of the 16-cell is and its vertex figure is a regular octahedron. There are 8 tetrahedra,12 triangles, and 6 edges meeting at every vertex and its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge, the 16-cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix and this decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell, or, Schläfli symbol ⨂ or ss, symmetry, order 64. The 16-cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16-cell center, one can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol, hence, the 16-cell has a dihedral angle of 120°. The dual tessellation, 24-cell honeycomb, is made of by regular 24-cells, together with the tesseractic honeycomb, these are the only three regular tessellations of R4. Each 16-cell has 16 neighbors with which it shares a tetrahedron,24 neighbors with which it only an edge. Twenty-four 16-cells meet at any vertex in this tessellation. A 16-cell can constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4-dimensional ring, the 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16-cell, the cell-first parallel projection of the 16-cell into 3-space has a cubical envelope
14.
5-cell
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In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid and it is a 4-simplex, the simplest possible convex regular 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base, the regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol. Pentachoron 4-simplex Pentatope Pentahedroid Pen Hyperpyramid, tetrahedral pyramid The 5-cell is self-dual and its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1, or approximately 75. 52°, the 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. The simplest set of coordinates is, with edge length 2√2, a 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, the purple edges represent the Petrie polygon of the 5-cell. The A4 Coxeter plane projects the 5-cell into a regular pentagon, the four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral pyramid vertex figures, Other uniform 5-polytopes have irregular 5-cell vertex figures, the symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and this compound has symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell, the pentachoron is the simplest of 9 uniform polychora constructed from the Coxeter group. It is in the sequence of regular polychora, the tesseract, 120-cell, of Euclidean 4-space, all of these have a tetrahedral vertex figure. It is similar to three regular polychora, the tesseract, 600-cell of Euclidean 4-space, and the order-6 tetrahedral honeycomb of hyperbolic space, all of these have a tetrahedral cell. T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
15.
Tetrahedron
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In geometry, a tetrahedron, 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, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
16.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
17.
Rectified 5-simplexes
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In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex, vertices of the birectified 5-simplex are located in the triangular face centers of the 5-simplex. In five dimensional geometry, a rectified 5-simplex, is a uniform 5-polytope with 15 vertices,60 edges,80 triangular faces,45 cells and it is also called 03,1 for its branching Coxeter-Dynkin diagram, shown as. E. L. Elte identified it in 1912 as a semiregular polytope, the rectified 5-simplex,031, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 13k series. The fifth figure is a Euclidean honeycomb,331, and the final is a noncompact hyperbolic honeycomb,431, each progressive uniform polytope is constructed from the previous as its vertex figure. Rectified hexateron The vertices of the rectified 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of or and these construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively. The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells and it has 20 vertices,90 edges,120 triangular faces,60 cells. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 02,2 for its branching Coxeter-Dynkin diagram, shown as. It is seen in the figure of the 6-dimensional 122. Birectified hexateron dodecateron The A5 projection has an appearance to Metatrons Cube. The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration, the vertices of a birectification exist at the center of the faces of the original polytope. It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cubes long diagonal orthogonally, in this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell. This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space, the vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of. This construction can be seen as facets of the birectified 6-orthoplex, the birectified 5-simplex,022, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The birectified 5-simplex is the figure for the third, the 122. The fourth figure is a Euclidean honeycomb,222, and the final is a noncompact hyperbolic honeycomb,322, each progressive uniform polytope is constructed from the previous as its vertex figure. This polytope is the figure of the 6-demicube, and the edge figure of the uniform 231 polytope
18.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21
19.
Dodecagon
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In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=
20.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators
21.
Convex polytope
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A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms polytope and convex polyhedron interchangeably. In addition, some require a polytope to be a bounded set. The terms bounded/unbounded convex polytope will be used whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or -manifold, Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum, in 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaums book, and in other texts in discrete geometry. Grünbaum points out that this is solely to avoid the repetition of the word convex. A polytope is called if it is an n-dimensional object in Rn. Many examples of bounded convex polytopes can be found in the article polyhedron, a convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaums definition is in terms of a set of points in space. Other important definitions are, as the intersection of half-spaces and as the hull of a set of points. This is equivalent to defining a bounded convex polytope as the hull of a finite set of points. Such a definition is called a vertex representation, for a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope may be defined as an intersection of a number of half-spaces. Such definition is called a half-space representation, there exist infinitely many H-descriptions of a convex polytope. However, for a convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as an inequality, a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b where n is the dimension of the space containing the polytope under consideration
22.
Isohedral figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
23.
Simple Lie group
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Simple Lie groups are a class of Lie groups which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups. Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative. Many commonly encountered Lie groups are simple or close to being simple, for example. In group theory, a simple Lie group is a locally compact non-abelian Lie group G which does not have nontrivial connected normal subgroups. A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0, an equivalent definition of a simple Lie group follows from the Lie correspondence, a connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups and it emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for special examples and configurations in other branches of mathematics. All Lie groups are smooth manifolds, mathematicians often study complex Lie groups, which are Lie groups with a complex structure on the underlying manifold, which is required to be compatible with the group operations. A complex Lie group is called if it is connected as a topological space. Note that the underlying Lie group may not be simple, although it still be semisimple. It is often useful to study slightly more general classes of Lie groups than simple groups, namely semisimple or, more generally, reductive Lie groups. A connected Lie group is called if its Lie algebra is a semisimple lie algebra. It is called if its Lie algebra is a direct sum of simple. Reductive groups occur naturally as symmetries of a number of objects in algebra, geometry. For example, the group G L n of symmetries of a real vector space is reductive. Finite-dimensional representations of simple groups split into direct sums of irreducible representations, simple Lie groups are fully classified. The classification is usually stated in several steps, namely, Classification of simple complex Lie algebras The classification of simple Lie algebras over the numbers by Dynkin diagrams. Classification of centerless Lie groups For every simple Lie algebra g, there is a unique centerless simple Lie group G whose Lie algebra is g, Classification of simple Lie groups One can show that the fundamental group of any Lie group is a discrete commutative group
24.
Wythoff construction
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In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoffs kaleidoscopic construction and it is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope, however, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point, if the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a point inside the spherical triangle enclosed by the mirrors. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron and this produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and similarly for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C and this produces a polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC and this produces a polyhedron with Wythoff symbol a b c|. The vertex is at a point such that, when it is rotated around any of the corners by twice the angle at that point. Only even-numbered reflections of the vertex are used. The polyhedron has the Wythoff symbol |a b c, the process in general also applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes. Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian and they generally can be derived from Wythoffian forms either by alternation or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry, sometimes snub forms are considered Wythoffian, even though they can only be constructed by the alternation of omnitruncated forms. Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra, coxeter-Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs. Coxeter Regular Polytopes, Third edition, Dover edition, ISBN 0-486-61480-8 Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 HarEl, Z. W. A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences,20 966–970, archived from the original on 4 February 2007. Displays Uniform Polyhedra using Wythoffs construction method Description of Wythoff Constructions Jenn, software that generates views of polyhedra and polychora from symmetry groups
25.
Birectified 5-cell
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In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra, each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces,30 edges, and 10 vertices, each vertex is surrounded by 3 octahedra and 2 tetrahedra, the vertex figure is a triangular prism. The vertex figure of the rectified 5-cell is a triangular prism. Together with the simplex and 24-cell, this shape and its dual was one of the first 2-simple 2-simplicial 4-polytopes known and this means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another pair of examples. The birectified 5-cell can be seen as the intersection of two regular 5-cells in dual positions and it is one of three semiregular 4-polytope made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5. These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively and this polytope is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope. It is also one of 9 Uniform 4-polytopes constructed from the Coxeter group, the rectified 5-cell is second in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed as the figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, the Coxeter symbol for the rectified 5-cell is 021. Semiregular k 21 polytope T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Rectified 5-cell - data, convex uniform polychora based on the pentachoron - Model 2, George Olshevsky
26.
3-3 duoprism
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In geometry of 4 dimensions, a 3-3 duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles. It has 9 vertices,18 edges,15 faces, in 6 triangular prism cells and it has Coxeter diagram, and symmetry, order 72. There are three constructions for the honeycomb with two lower symmetries, the regular complex polytope 32, in C2 has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, and 6 3-edges and its symmetry is 32, order 18. It also has a lower construction, or 3×3, with symmetry 33. This is the if the red and blue 3-edges are considered distinct. The dual of a 3-3 duoprism is called a 3-3 duopyramid and it has 9 tetragonal disphenoid cells,18 triangular faces,15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, orthogonal projection The regular complex polygon 23 has 6 vertices in C2 with a real represention in R4 matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the edges of the 3-3 duopyramid. It can be seen in a projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage, 3-4 duoprism Tesseract 5-5 duoprism Convex regular 4-polytope Duocylinder Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
27.
Disphenoid
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In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which two edges that are opposite each other have equal lengths. Other names for the shape are sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron. All the solid angles and vertex figures of a disphenoid are the same, however, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths. If the faces of a disphenoid are equilateral triangles, it is a tetrahedron with Td tetrahedral symmetry. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid, in this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid, unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. Both tetragonal disphenoids and rhombic disphenoids are isohedra, as well as being congruent to each other and it is not possible to construct a disphenoid with right triangle or obtuse triangle faces. When right triangles are glued together in the pattern of a disphenoid, two more types of tetrahedron generalize the disphenoid and have similar names. The digonal disphenoid has faces with two different shapes, both triangles, with two faces of each shape. The phyllic disphenoid similarly has faces with two shapes of scalene triangles, disphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms. A tetrahedron is a if and only if its circumscribed parallelepiped is right-angled. We also have that a tetrahedron is a if and only if the center in the circumscribed sphere. The disphenoids are the polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting and they are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints. The volume of a disphenoid with opposite edges of length l, m and n is given by V =72. There is also the following interesting relation connecting the volume and the circumradius,16 T2 R2 = l 2 m 2 n 2 +9 V2, the squares of the lengths of the bimedians are 12,12,12. If the four faces of a tetrahedron have the same perimeter, if the four faces of a tetrahedron have the same area, then it is a disphenoid
28.
Isosceles triangle
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
29.
Complex polytope
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In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes. Precise definitions exist only for the regular polytopes, which are configurations. The regular complex polytopes have been characterized, and can be described using a symbolic notation developed by Coxeter. Some complex polytopes which are not fully regular have also been described, the complex line C1 has one dimension with real coordinates and another with imaginary coordinates. Applying real coordinates to both dimensions is said to give it two dimensions over the real numbers, a real plane, with the imaginary axis labelled as such, is called an Argand diagram. Because of this it is called the complex plane. Complex 2-space is thus a four-dimensional space over the reals, a complex n-polytope in complex n-space is the analogue of a real n-polytope in real n-space. There is no natural complex analogue of the ordering of points on a real line, because of this a complex polytope cannot be seen as a contiguous surface and it does not bound an interior in the way that a real polytope does. In the case of polytopes, a precise definition can be made by using the notion of symmetry. For any regular polytope the symmetry group acts transitively on the flags, thus, by definition, regular complex polytopes are configurations in complex unitary space. The regular complex polytopes were discovered by Shephard, and the theory was developed by Coxeter. A complex polytope exists in the space of equivalent dimension. For example, the vertices of a polygon are points in the complex plane C2. Thus, an edge can be given a system consisting of a single complex number. In a regular polytope the vertices incident on the edge are arranged symmetrically about their centroid. So we may assume that the vertices on the edge satisfy the equation x p −1 =0 where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the points lie at the vertices of a regular polygon centered on the origin
30.
Rectification (geometry)
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In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the facets of the original polytope. A rectification operator is denoted by the symbol r, for example, r is the rectified cube. Conway polyhedron notation uses ambo for this operator, in graph theory this operation creates a medial graph. Rectification is the point of a truncation process. The highest degree of rectification creates the dual polytope, a rectification truncates edges to points. A birectification truncates faces to points, a trirectification truncates cells to points, and so on. New vertices are placed at the center of the edges of the original polygon, each platonic solid and its dual have the same rectified polyhedron. The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual, the rectified octahedron, whose dual is the cube, is the cuboctahedron. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron, a rectified square tiling is a square tiling. A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling, examples If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. The resulting medial graph remains polyhedral, so by Steinitzs theorem it can be represented as a polyhedron, the Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, is Conways expand operation, e, which is the same as Johnsons cantellation operation, t0,2 generated from regular polyhedral, each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. Its rectification will have two types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex. A rectified is not the same as a rectified, however, a further truncation, called bitruncation, is symmetric between a 4-polytope and its dual. Examples A first rectification truncates edges down to points, If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1 or r. A second rectification, or birectification, truncates faces down to points, If regular it has notation t2 or 2r. For polyhedra, a birectification creates a dual polyhedron, higher degree rectifications can be constructed for higher dimensional polytopes
31.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
32.
Four-dimensional space
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For example, the volume of a rectangular box is found by measuring its length, width, and depth. More than two millennia ago Greek philosophers explored in detail the implications of this uniformity, culminating in Euclids Elements. However, it was not until recent times that a handful of insightful mathematical innovators generalized the concept of dimensions to more than three. The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s, in 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension. Which was notable for explaining the concept of a cube by going through a step-by-step generalization of the properties of lines, squares. The simplest form of Hintons method is to draw two ordinary cubes separated by a distance, and then draw lines between their equivalent vertices. This form can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube, the eight lines connecting the vertices of the two cubes in that case represent a single direction in the unseen fourth dimension. Higher dimensional spaces have become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces, calendar entries for example are usually 4D locations, such as a meeting at time t at the intersection of two streets on some building floor. In list form such a meeting place at the 4D location. Einsteins concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space, when dimensional locations are given as ordered lists of numbers such as they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the richness and geometric complexity of 4D. A hint of that complexity can be seen in the animation of one of simplest possible 4D objects. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time, the possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843 and this associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R, one of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension. Published in the Dublin University magazine and he coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension. Hintons ideas inspired a fantasy about a Church of the Fourth Dimension featured by Martin Gardner in his January 1962 Mathematical Games column in Scientific American, in 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams
33.
Five-dimensional space
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A five-dimensional space is a space with five dimensions. If interpreted physically, that is one more than the three spatial dimensions and the fourth dimension of time used in relativitistic physics. It is an abstraction which occurs frequently in mathematics, where it is a legitimate construct, in physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. Whether or not the universe is five-dimensional is a topic of debate, 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 not be observable, Klein suggested that the fifth dimension would be rolled up into a tiny. While not detectable, it would imply a connection between seemingly unrelated forces. Superstring theory then evolved into a generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings, the other 10 dimensions are compacted, or rolled up, to a size below the subatomic level. Kaluza–Klein theory today is seen as essentially a gauge theory, with the gauge being the circle group, the fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct and these theories make reference to Hilbert space, a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. They suggested that electromagnetism resulted from a field that is “polarized” in the fifth dimension. The main novelty of Einstein and Bergmann was to consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor. But they then reneged, modifying the theory to break its five-dimensional symmetry, minkowski space and Maxwells equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal particle. T Hooft has speculated that the dimension is really the spacetime fabric. According to Klein’s definition, a geometry is the study of the invariant properties of a spacetime, therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. In five or more dimensions, only three regular polytopes exist, in five dimensions, they are, The 5-simplex of the simplex family, with 6 vertices,15 edges,20 faces,15 cells, and 6 hypercells. The 5-cube of the family, with 32 vertices,80 edges,80 faces,40 cells
34.
Six-dimensional space
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Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are a number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes, six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space, as such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined, and can be used to calculate the metric,6 ×6 matrices can be used to describe transformations such as rotations that keep the origin fixed. More generally, any space that can be described locally with six coordinates, one example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidean space ℝ7 that are equidistant from the origin and this constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, a polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are three in six dimensions, the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 6-demicube is a unique polytope from the D6 family, and 221 and 122 polytopes from the E6 family. The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point and it has symbol S5, and the equation for the 5-sphere, radius r, centre the origin is S5 =. The volume of space bounded by this 5-sphere is V6 = π3 r 66 which is 5.16771 × r6. The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point and it has symbol S6, and the equation for the 6-sphere, radius r, centre the origin is S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3 r 7105 which is 4.72477 × r7. In three dimensional space a transformation has six degrees of freedom, three translations along the three coordinate axes and three from the rotation group SO. Often these transformations are handled separately as they have different geometrical structures. In screw theory angular and linear velocity are combined into one six-dimensional object, a similar object called a wrench combines forces and torques in six dimensions
35.
Seven-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n =7, the set of all locations is called 7-dimensional space. Often such a space is studied as a space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, more generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a manifold such as a 7-sphere. Seven-dimensional spaces have a number of properties, many of them related to the octonions. An especially distinctive property is that a product can be defined only in three or seven dimensions. This is related to Hurwitzs theorem, which prohibits the existence of structures like the quaternions and octonions in dimensions other than 2,4. The first exotic spheres ever discovered were seven-dimensional, a polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are three in seven dimensions, the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 7-demicube is a unique polytope from the D7 family, and 321,231, and 132 polytopes from the E7 family. The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a point and it has symbol S6, with formal definition for the 6-sphere with radius r of S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3105 r 7 which is 4.72477 × r7. A cross product, that is a valued, bilinear, anticommutative. Along with the usual cross product in three dimensions it is the only such product, except for trivial products. In 1956, John Milnor constructed an exotic sphere in 7 dimensions, in 1963 he showed that the exact number of such structures is 28. Euclidean geometry List of geometry topics List of regular polytopes H. S. M, dover,1973 J. W. Milnor, On manifolds homeomorphic to the 7-sphere
36.
Eight-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n =8, the set of all locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance, eight-dimensional Euclidean space is eight-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally the term may refer to a vector space over any field, such as an eight-dimensional complex vector space. It may also refer to a manifold such as an 8-sphere. A polytope in eight dimensions is called an 8-polytope, the most studied are the regular polytopes, of which there are only three in eight dimensions, the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 8-demicube is a unique polytope from the D8 family, and 421,241, and 142 polytopes from the E8 family. The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point and it has symbol S7, with formal definition for the 7-sphere with radius r of S7 =. The volume of the bounded by this 7-sphere is V8 = π424 R8 which is 4.05871 × r8. The kissing number problem has been solved in eight dimensions, thanks to the existence of the 421 polytope, the kissing number in eight dimensions is 240. The octonions are a division algebra over the real numbers. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, a normed algebra is one with a product that satisfies ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ for all x and y in the algebra. A normed division algebra additionally must be finite-dimensional, and have the property that every non-zero vector has a multiplicative inverse. Hurwitzs theorem prohibits such a structure from existing in other than 1,2,4. The complexified quaternions C ⊗ H, or biquaternions, are an eight-dimensional algebra dating to William Rowan Hamiltons work in the 1850s and this algebra is equivalent to the Clifford algebra C ℓ2 and the Pauli algebra. It has also proposed as a practical or pedagogical tool for doing calculations in special relativity. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C
37.
Nine-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a point in n-dimensional space. When n =9, the set of all locations is called 9-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance, nine-dimensional Euclidean space is nine-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally, the term may refer to a vector space over any field, such as a nine-dimensional complex vector space. It may also refer to a manifold such as a 9-sphere. A polytope in nine dimensions is called an 9-polytope, the most studied are the regular polytopes, of which there are only three in nine dimensions, the 9-simplex, 9-cube, and 9-orthoplex. A broader family are the uniform 9-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 9-demicube is a unique polytope from the D9 family. H. S. M. Coxeter, H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 Wiley, Kaleidoscopes, Selected Writings of H. S. M
38.
String theory
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In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how strings propagate through space and interact with each other. On distance scales larger than the scale, a string looks just like an ordinary particle, with its mass, charge. In string theory, one of the vibrational states of the string corresponds to the graviton. Thus string theory is a theory of quantum gravity, String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. Despite much work on problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows to choose the details. String theory was first studied in the late 1960s as a theory of the nuclear force. Subsequently, it was realized that the properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, one of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and these issues have led some in the community to criticize these approaches to physics and question the value of continued research on string theory unification. In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics, one of these frameworks was Albert Einsteins general theory of relativity, a theory that explains the force of gravity and the structure of space and time. The other was quantum mechanics, a different formalism for describing physical phenomena using probability. In spite of successes, there are still many problems that remain to be solved. One of the deepest problems in physics is the problem of quantum gravity. The general theory of relativity is formulated within the framework of classical physics, in addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, and the early universe. String theory is a framework that attempts to address these questions
39.
E7 (mathematics)
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The E7 algebra is thus one of the five exceptional cases. The fundamental group of the form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z. The dimension of its representation is 56. There is a unique complex Lie algebra of type E7, corresponding to a group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form of E7, the split form, EV, which has maximal compact subgroup SU/, fundamental group cyclic of order 4 and outer automorphism group of order 2. EVI, which has maximal compact subgroup SU·SO/, fundamental group non-cyclic of order 4, EVII, which has maximal compact subgroup SO·E6/, infinite cyclic findamental group and outer automorphism group of order 2. For a complete list of forms of simple Lie algebras. The compact real form of E7 is the group of the 64-dimensional exceptional compact Riemannian symmetric space EVI. This can be seen using a construction known as the magic square, due to Hans Freudenthal. The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, over finite fields, the Lang–Steinberg theorem implies that H1 =0, meaning that E7 has no twisted forms, see below. The Dynkin diagram for E7 is given by, even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7-dimensional subspace of an 8-dimensional vector space. The roots are all the 8×7 permutations of and all the permutations of Note that the 7-dimensional subspace is the subspace where the sum of all the eight coordinates is zero. The simple roots are We have ordered them so that their corresponding nodes in the Dynkin diagram are ordered left to right with the side node last. Given the E7 Cartan matrix and a Dynkin diagram node ordering of, the Weyl group of E7 is of order 2903040, it is the direct product of the cyclic group of order 2 and the unique simple group of order 1451520. E7 has an SU subalgebra, as is evident by noting that in the 8-dimensional description of the root system, in addition to the 133-dimensional adjoint representation, there is a 56-dimensional vector representation, to be found in the E8 adjoint representation. The characters of finite dimensional representations of the real and complex Lie algebras, there exist non-isomorphic irreducible representation of dimensions 1903725824,16349520330, etc. The fundamental representations are those with dimensions 133,8645,365750,27664,1539,56 and 912, E7 is the automorphism group of the following pair of polynomials in 56 non-commutative variables