1.
Projection (linear algebra)
–
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.
Petrie polygon
–
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
3.
Uniform 8-polytope
–
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
4.
Simplex
–
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices, a regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex, the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨
5.
7-simplex
–
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices,28 edges,56 triangle faces,70 tetrahedral cells,56 5-cell 5-faces,28 5-simplex 6-faces and its dihedral angle is cos−1, or approximately 81. 79°. It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions, the name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca, the Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are, More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of. This construction is based on facets of the 8-orthoplex and this polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram, This polytope is one of 71 uniform 7-polytopes with A7 symmetry. Polytopes of Various Dimensions Multi-dimensional Glossary
6.
6-simplex
–
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices,21 edges,35 triangle faces,35 tetrahedral cells,21 5-cell 4-faces and its dihedral angle is cos−1, or approximately 80. 41°. It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions, the name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop, the regular 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 Coxeter plane orthographic projections. 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. 6D uniform polytopes x3o3o3o3o - hix, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
7.
5-simplex
–
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices,15 edges,20 triangle faces,15 tetrahedral cells and it has a dihedral angle of cos−1, or approximately 78. 46°. It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions, the name hexateron is derived from hexa- for having six facets and teron for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix, the hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell. These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron, the 5-simplex, as 220 polytope is first in dimensional series 22k. The regular 5-simplex is one of 19 uniform polytera based on the Coxeter group, the 5-simplex can also be considered a 5-cell pyramid, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells, 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. 5D uniform polytopes x3o3o3o3o - hix, archived from the original on 4 February 2007. Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary
8.
5-cell
–
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