1.
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
2.
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
3.
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
4.
Uniform 8-polytope
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In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets, a uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes can be represented by the Schläfli symbol, with v 7-polytope facets around each peak, There are exactly three such convex regular 8-polytopes, - 8-simplex - 8-cube - 8-orthoplex There are no nonconvex regular 8-polytopes. The topology of any given 8-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Bowers-style acronym names are given in parentheses for cross-referencing, see also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes. The B8 family has symmetry of order 10321920, There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes, the D8 family has symmetry of order 5,160,960. This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D8 Coxeter-Dynkin diagram with one or more rings,127 are repeated from the B8 family and 64 are unique to this family, all listed below. See list of D8 polytopes for Coxeter plane graphs of these polytopes, the E8 family has symmetry order 696,729,600. There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, eight forms are shown below,4 single-ringed,3 truncations, and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing, see also list of E8 polytopes for Coxeter plane graphs of this family. However, there are 4 noncompact hyperbolic Coxeter groups of rank 8, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 A. S. M. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, 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 Wiley, Kaleidoscopes, Selected Writings of 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, N. W, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
5.
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
6.
8-simplex
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In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices,36 edges,84 triangle faces,126 tetrahedral cells,126 5-cell 4-faces,84 5-simplex 5-faces,36 6-simplex 6-faces and its dihedral angle is cos−1, or approximately 82. 82°. It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions, the name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on. This construction is based on facets of the 9-orthoplex and this polytope is a facet in the uniform tessellations,251, and 521 with respective Coxeter-Dynkin diagrams, This polytope is one of 135 uniform 8-polytopes with A8 symmetry. 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. 8D uniform polytopes x3o3o3o3o3o3o3o - ene, Polytopes of Various Dimensions Multi-dimensional Glossary
7.
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