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.
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
3.
Uniform 10-polytope
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In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets, Regular 10-polytopes can be represented by the Schläfli symbol, with x 9-polytope facets around each peak. There are exactly three convex regular 10-polytopes, - 10-simplex - 10-cube - 10-orthoplex There are no nonconvex regular 10-polytopes. The topology of any given 10-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 10-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 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below, all one and two ringed forms, and the final omnitruncated form, bowers-style acronym names are given in parentheses for cross-referencing. There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, twelve cases are shown below, ten single-ring forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing, the D10 family has symmetry of order 1,857,945,600. This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram, of these,511 are repeated from the B10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing, however, there are 3 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams. 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 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. Dissertation, University of Toronto,1966 Klitzing, Richard, polytope names Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary Glossary for hyperspace, George Olshevsky
4.
Hypercube
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In geometry, a hypercube is an n-dimensional analogue of a square and a cube. A unit hypercubes longest diagonal in n-dimensions is equal to n, an n-dimensional hypercube is also called an n-cube or an n-dimensional cube. The term measure polytope is also used, notably in the work of H. S. M. Coxeter, the hypercube is the special case of a hyperrectangle. A unit hypercube is a hypercube whose side has one unit. Often, the hypercube whose corners are the 2n points in Rn with coordinates equal to 0 or 1 is called the unit hypercube, a hypercube can be defined by increasing the numbers of dimensions of a shape,0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment,2 – If one moves this line segment its length in a perpendicular direction from itself, it sweeps out a 2-dimensional square. 3 – If one moves the square one unit length in the perpendicular to the plane it lies on. 4 – If one moves the cube one unit length into the fourth dimension and this can be generalized to any number of dimensions. The 1-skeleton of a hypercube is a hypercube graph, a unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates. It has a length of 1 and an n-dimensional volume of 1. An n-dimensional hypercube is also regarded as the convex hull of all sign permutations of the coordinates. This form is chosen due to ease of writing out the coordinates. Its edge length is 2, and its volume is 2n. Every n-cube of n >0 is composed of elements, or n-cubes of a dimension, on the -dimensional surface on the parent hypercube. A side is any element of -dimension of the parent hypercube, a hypercube of dimension n has 2n sides. The number of vertices of a hypercube is 2 n, the number of m-dimensional hypercubes on the boundary of an n-cube is E m, n =2 n − m, where = n. m. and n. denotes the factorial of n. For example, the boundary of a 4-cube contains 8 cubes,24 squares,32 lines and 16 vertices and this identity can be proved by combinatorial arguments, each of the 2 n vertices defines a vertex in a m-dimensional boundary. There are ways of choosing which lines that defines the subspace that the boundary is in, but, each side is counted 2 m times since it has that many vertices, we need to divide with this number
5.
9-cube
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It can be named by its Schläfli symbol, being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract and it can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets. It is a part of an family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the family of cross-polytopes. Cartesian coordinates for the vertices of a 9-cube centered at the origin, applying an alternation operation, deleting alternating vertices of the 9-cube, creates another uniform polytope, called a 9-demicube, which has 18 8-demicube and 256 8-simplex facets. 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, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, 9D uniform polytopes o3o3o3o3o3o3o3o4x - enne. Archived from the original on 4 February 2007
6.
8-cube
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In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices,1024 edges,1792 square faces,1792 cubic cells,1120 tesseract 4-faces,448 5-cube 5-faces,112 6-cube 6-faces and it is represented by Schläfli symbol, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract and oct for eight in Greek and it can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets. It is a part of an family of polytopes, called hypercubes. The dual of an 8-cube can be called a 8-orthoplex, and is a part of the family of cross-polytopes. Cartesian coordinates for the vertices of an 8-cube centered at the origin, applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube, which has 16 demihepteractic and 128 8-simplex facets. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, 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, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, 8D uniform polytopes o3o3o3o3o3o3o4x - octo. Archived from the original on 4 February 2007
7.
7-cube
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In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices,448 edges,672 square faces,560 cubic cells,280 tesseract 4-faces,84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol, being composed of 3 6-cubes around each 5-face and it can be called a hepteract, a portmanteau of tesseract and hepta for seven in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets and it is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube is called a 7-orthoplex, and is a part of the family of cross-polytopes. Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, which has 14 demihexeractic and 64 6-simplex 6-faces. Cartesian coordinates for the vertices of a hepteract centered at the origin, hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D. 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, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, 7D uniform polytopes o3o3o3o3o3o4x - hept. Archived from the original on 4 February 2007, multi-dimensional Glossary, hypercube Garrett Jones Rotation of 7D-Cube www. 4d-screen. de
8.
6-cube
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In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices,192 edges,240 square faces,160 cubic cells,60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol, being composed of 3 5-cubes around each 4-face and it can be called a hexeract, a portmanteau of tesseract with hex for six in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets and it is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the family of cross-polytopes. Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, which has 12 5-demicube and 32 5-simplex facets. Cartesian coordinates for the vertices of a 6-cube centered at the origin and this polytope is one of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p.296, Table I, Regular Polytopes, 6D uniform polytopes o3o3o3o3o4x - ax. Archived from the original on 4 February 2007