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 7-polytope
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In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets, a uniform 7-polytope is one which is vertex-transitive, and constructed from uniform 6-polytope facets. Regular 7-polytopes are represented by the Schläfli symbol with u 6-polytopes facets around each 4-face, There are exactly three such convex regular 7-polytopes, - 7-simplex - 7-cube - 7-orthoplex There are no nonconvex regular 7-polytopes. The topology of any given 7-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, 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 71 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Norman Johnsons truncation names are given, bowers names and acronym are also given for cross-referencing. See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes, the B7 family has symmetry of order 645120. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, see also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes. The D7 family has symmetry of order 322560 and this family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these,63 are repeated from the B7 family and 32 are unique to this family, bowers names and acronym are given for cross-referencing. See also list of D7 polytopes for Coxeter plane graphs of these polytopes, the E7 Coxeter group has order 2,903,040. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, see also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes. Coxeter calls the first one a quarter 6-cubic honeycomb, however, there are 3 noncompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams. The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a Coxeter-Dynkin diagram, an active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope, Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways, here are the primary operators available for constructing and naming the uniform 7-polytopes. The prismatic forms and bifurcating graphs can use the same truncation indexing notation, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 A. S. M
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.
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
6.
5-cube
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In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices,80 edges,80 square faces,40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or, constructed as 3 tesseracts and it can be called a penteract, a portmanteau of tesseract and pente for five in Greek. It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets and it is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the family of orthoplexes. The 5-cube can be seen as an order-3 tesseractic honeycomb on a 4-sphere and it is related to the Euclidean 4-space tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic honeycomb. This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex. 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, 5D uniform polytopes o3o3o3o4x - pent. Archived from the original on 4 February 2007
7.
Tesseract
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In geometry, the tesseract is the four-dimensional analog of the cube, the tesseract is to the cube as the cube is to the square. Just as the surface of the consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384, constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64, as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
8.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
9.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
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.
6-simplex
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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
12.
Tetradecagon
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In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. A regular tetradecagon has Schläfli symbol and can be constructed as a quasiregular truncated heptagon, t, however, it is constructible using neusis, as for example in the following illustration with use of the angle trisector. Based on the circle r =1 Constructed side length of the tetradecagon in GeoGebra a =0.445041867912629. Side length of the tetradecagon a s h o u l d =2 ⋅ sin =0.4450418679126288089, for details, see, Wikibooks, Tetradecagon, construction description The regular tetradecagon has Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries, Dih7, Dih2, and Dih1, and 4 cyclic group symmetries, Z14, Z7, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the tetradecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r28 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g14 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular tetradecagon. The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, a tetradecagram is a 14-sided star polygon, represented by symbol. There are two regular polygons, and, using the same vertices, but connecting every third or fifth points. There are also three compounds, is reduced to 2 as two heptagons, while and are reduced to 2 and 2 as two different heptagrams, and finally is reduced to seven digons. Deeper truncations of the regular heptagon and heptagrams can produce isogonal intermediate tetradecagram forms with equally spaced vertices, other truncations can form double covering polyons 2, namely, t==2, t==2, and t==2. Regular skew tetradecagons exist as Petrie polygon for many higher-dimensional polytopes, shown in these orthogonal projections, including, Weisstein
13.
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
14.
7-orthoplex
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In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices,84 edges,280 triangle faces,560 tetrahedron cells,672 5-cells 4-faces,448 5-faces, and 128 6-faces. It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets and it is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract, heptacross, derived from combining the family name cross polytope with hept for seven in Greek. A lowest symmetry construction is based on a dual of a 7-orthotope, cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are, Every vertex pair is connected by an edge, except opposites. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, 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 x3o3o3o3o3o4o - zee. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
15.
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
16.
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
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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
18.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
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Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
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Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
21.
5-face
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
22.
6-face
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
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Portmanteau
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In linguistics, a portmanteau is defined as a single morph that represents two or more morphemes. A portmanteau also differs from a compound, which not involve the truncation of parts of the stems of the blended words. For instance, starfish is a compound, not a portmanteau, of star and fish, whereas a hypothetical portmanteau of star and fish might be stish. Humpty Dumpty explains the practice of combining words in various ways by telling Alice, for instance, take the two words fuming and furious. Make up your mind that you will say both words, but leave it unsettled which you will say first … if you have the rarest of gifts, in then-contemporary English, a portmanteau was a suitcase that opened into two equal sections. The etymology of the word is the French porte-manteau, from porter, to carry, in modern French, a porte-manteau is a clothes valet, a coat-tree or similar article of furniture for hanging up jackets, hats, umbrellas and the like. It has also used especially in Europe as a formal description for hat racks from the French words porter. An occasional synonym for portmanteau word is frankenword, an autological word exemplifying the phenomenon it describes, blending Frankenstein, many neologisms are examples of blends, but many blends have become part of the lexicon. In Punch in 1896, the word brunch was introduced as a portmanteau word, in 1964, the newly independent African republic of Tanganyika and Zanzibar chose the portmanteau word Tanzania as its name. Similarly Eurasia is a portmanteau of Europe and Asia, a scientific example is a liger, which is a cross between a male lion and a female tiger. Jeoportmanteau. is a category on the American television quiz show Jeopardy. The categorys name is itself a portmanteau of the words Jeopardy, responses in the category are portmanteaus constructed by fitting two words together. The term gerrymander has itself contributed to portmanteau terms bjelkemander and playmander, oxbridge is a common portmanteau for the UKs two oldest universities, those of Oxford and Cambridge. Many portmanteau words receive some use but do not appear in all dictionaries, for example, a spork is an eating utensil that is a combination of a spoon and a fork, and a skort is an item of clothing that is part skirt, part shorts. On the other hand, turducken, a made by inserting a chicken into a duck. Similarly, the word refudiate was first used by Sarah Palin when she misspoke, though initially a gaffe, the word was recognized as the New Oxford American Dictionarys Word of the Year in 2010. The business lexicon is replete with newly coined portmanteau words like permalance, advertainment, advertorial, infotainment, a company name may also be portmanteau as well as a product name. By contrast, the public, including the media, use portmanteaux to refer to their favorite pairings as a way to. giv people an essence of who they are within the same name and this is particularly seen in cases of fictional and real-life supercouples
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Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
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Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
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Cross-polytope
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In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-orthoplex is a square, a 3-orthoplex is an octahedron. Its facets are simplexes of the dimension, while the cross-polytopes vertex figure is another cross-polytope from the previous dimension. The vertices of a cross-polytope are all the permutations of, the cross-polytope is the convex hull of its vertices. The n-dimensional cross-polytope can also be defined as the unit ball in the ℓ1-norm on Rn. In 1 dimension the cross-polytope is simply the line segment, in 2 dimensions it is a square with vertices, in 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these, the cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T, the 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of six convex regular 4-polytopes and these 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. The cross polytope family is one of three regular polytope families, labeled by Coxeter as βn, the two being the hypercube family, labeled as γn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn, the n-dimensional cross-polytope has 2n vertices, and 2n facets all of which are n−1 simplices. The vertex figures are all n −1 cross-polytopes, the Schläfli symbol of the cross-polytope is. The dihedral angle of the n-dimensional cross-polytope is δ n = arccos and this gives, δ2 = arccos = 90°, δ3 = arccos =109. 47°, δ4 = arccos = 120°, δ5 = arccos =126. 87°. The volume of the n-dimensional cross-polytope is 2 n n. Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons. A second projection takes the 2-gon petrie polygon of the dimension, seen as a bipyramid, projected down the axis. The vertices of a cross polytope are all at equal distance from each other in the Manhattan distance. Kusners conjecture states that this set of 2d points is the largest possible equidistant set for this distance, Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes, βpn =22. 2p, or. Real solutions exist with p=2, i. e. β2n = βn =22.22 =, for p>2, they exist in C n
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Alternation (geometry)
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In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for hemi or half, because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a. 2b. 2c is a.3. b.3. c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons, a snub can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces, all truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub and this alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the vertices will not in general create uniform facets. Examples, Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb, an alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell, 4-honeycombs, An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube, cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract. Coxeter also used the operator a, which contains both halves, so retains the original symmetry, for even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron, Norman Johnson extended the use of the altered operator a, b for blended, and c for converted, as, and respectively. The compound polyhedron, stellated octahedron can be represented by a, the star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a, and. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the free edges. A similar operation can truncate alternate vertices, rather than just removing them, below is a set of polyhedra that can be generated from the Catalan solids. These have two types of vertices which can be alternately truncated, truncating the higher order vertices and both vertex types produce these forms, Conway polyhedral notation Wythoff construction Coxeter, H. S. M
28.
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
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7-demicube
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In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract, there are 95 uniform polytopes with D6 symmetry,63 are shared by the B6 symmetry, and 32 are unique, 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. 7D uniform polytopes x3o3o *b3o3o3o3o - hesa, archived from the original on 4 February 2007
30.
Demihypercube
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In geometry, demihypercubes are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed, the 2n facets become 2n -demicubes, and 2n -simplex facets are formed in place of the deleted vertices. They have been named with a prefix to each hypercube name, demicube, demitesseract. The demicube is identical to the tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having regular facets. Higher forms dont have all regular facets but are all uniform polytopes, the vertices and edges of a demihypercube form two copies of the halved cube graph. Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above 3 and he called it a 5-ic semi-regular. It also exists within the semiregular k21 polytope family, the demihypercubes can be represented by extended Schläfli symbols of the form h as half the vertices of. The vertex figures of demihypercubes are rectified n-simplexes and they are represented by Coxeter-Dynkin diagrams of three constructive forms. Coxeter also labeled the third bifurcating diagrams as 1k1 representing the lengths of the 3 branches, an n-demicube, n greater than 2, has n*/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection. Facets, Dn, n-1 = n + 2n The symmetry group of the demihypercube is the Coxeter group D n, has order 2 n −1 n. and is an index 2 subgroup of the hyperoctahedral group. It is generated by permutations of the axes and reflections along pairs of coordinate axes. Constructions as alternated orthotopes have the topology, but can be stretched with different lengths in n-axes of symmetry. The rhombic disphenoid is the example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces, Coxeter, editied 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 III, Olshevsky, George. Archived from the original on 4 February 2007
31.
6-demicube
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In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches. It can named similarly by a 3-dimensional exponential Schläfli symbol or, cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract, with an odd number of plus signs. 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. It is also the second in a series of uniform polytopes and honeycombs. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb,134. 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. 6D uniform polytopes x3o3o *b3o3o3o – hax, archived from the original on 4 February 2007
32.
Simplicial complex
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In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts. Simplicial complexes should not be confused with the abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. A simplicial complex K is a set of simplices that satisfies the conditions,1. Any face of a simplex from K is also in K.2, the intersection of any two simplices σ1, σ2 ∈ K is either ∅ or a face of both σ1 and σ2. Note that the empty set is a face of every simplex, see also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k-complex K is a complex where the largest dimension of any simplex in K equals k. For instance, a simplicial 2-complex must contain at least one triangle, a pure or homogeneous simplicial k-complex K is a simplicial complex where every simplex of dimension less than k is a face of some simplex σ ∈ K of dimension exactly k. Informally, a pure 1-complex looks like its made of a bunch of lines, an example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. A facet is any simplex in a complex that is not a face of any larger simplex, a pure simplicial complex can be thought of as a complex where all facets have the same dimension. Sometimes the term face is used to refer to a simplex of a complex, for a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is used in a broader sense to denote a set homeomorphic to a simplex. The underlying space, sometimes called the carrier of a complex is the union of its simplices. Let K be a complex and let S be a collection of simplices in K. The closure of S is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S is obtained by adding to S each face of every simplex in S. The star of S is the union of the stars of each simplex in S, for a single simplex s, the star of s is the set of simplices having a face in s. The link of S equals Cl St S − St Cl S and it is the closed star of S minus the stars of all faces of S. In algebraic topology, simplicial complexes are useful for concrete calculations
33.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
34.
Pascal's triangle
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In mathematics, Pascals triangle is a triangular array of the binomial coefficients. In the Western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, the rows of Pascals triangle are conventionally enumerated starting with row n =0 at the top. The entries in each row are numbered from the beginning with k =0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the manner, In row 0. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the number in the first row is 1. The entry in the nth row and kth column of Pascals triangle is denoted, for example, the unique nonzero entry in the topmost row is =1. With this notation, the construction of the previous paragraph may be written as follows, = +, for any integer n. This recurrence for the coefficients is known as Pascals rule. Pascals triangle has higher dimensional generalizations, the three-dimensional version is called Pascals pyramid or Pascals tetrahedron, while the general versions are called Pascals simplices. The pattern of numbers that forms Pascals triangle was known well before Pascals time, centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greeks study of figurate numbers. From later commentary, it appears that the coefficients and the additive formula for generating them. Halayudha also explained obscure references to Meru-prastaara, the Staircase of Mount Meru, in approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula = n. r. At around the time, it was discussed in Persia by the Persian mathematician. It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám, thus the triangle is referred to as the Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem, Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. Pascals triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian, in the 13th century, Yang Hui presented the triangle and hence it is still called Yang Huis triangle in China. In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, petrus Apianus published the full triangle on the frontispiece of his book on business calculations in 1527
35.
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
36.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
37.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
38.
Regular Polytopes (book)
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Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter. Originally published in 1947, the book was updated and republished in 1963 and 1973, the book is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of the book took Coxeter twenty-four years to complete, regular Polytopes is a standard reference work on regular polygons, polyhedra and their higher dimensional analogues. It is unusual in the breadth of its coverage, its combination of mathematical rigour with geometric insight, Coxeter starts by introducing two-dimensional polygons and three-dimensional polyhedra. He then gives a rigorous definition of regularity and uses it to show that there are no other convex regular polyhedra apart from the five Platonic solids. The concept of regularity is extended to non-convex shapes such as star polygons and star polyhedra, to tessellations and honeycombs, Coxeter introduces and uses the groups generated by reflections that became known as Coxeter groups. The book combines algebraic rigour with clear explanations, many of which are illustrated with diagrams, the black and white plates in the book show solid models of three-dimensional polyhedra, and wire-frame models of projections of some higher-dimensional polytopes. At the end of each chapter Coxeter includes an Historical remarks section which provides a perspective of the development of the subject. The first two have been expounded by Sommerville and Neville, and we shall presuppose some familiarity with such treatises. Concerning the third, Poincaré wrote, A man who pursues it, will end up holding on to the fourth dimension. ”The original 1948 edition received a more complete review by M. Goldberg in MR0027148
39.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
40.
Norman Johnson (mathematician)
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Norman Woodason Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. He earned his Ph. D. from the University of Toronto in 1966 with a title of The Theory of Uniform Polytopes. In 1966 he enumerated 92 convex non-uniform polyhedra with regular faces, victor Zalgaller later proved that Johnsons list was complete, and the set is now known as the Johnson solids. The theory of polytopes and honeycombs, Ph. D. Dissertation,1966 Hyperbolic Coxeter Groups, paper, convex polyhedra with regular faces, paper containing the original enumeration of the 92 Johnson solids and the conjecture that there are no others. Norman W. Johnson at the Mathematics Genealogy Project Norman W. Johnson Endowed Fund in Mathematics and Computer Science at Wheaton College
41.
Regular polytope
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike, note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form, with regular facets as, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object, some of these have regular examples, as discussed in the section on historical discovery below. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, the notation is best explained by adding one dimension at a time. A convex regular polygon having n sides is denoted by, so an equilateral triangle is, a square, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the figure of the polyhedron. A regular 4-polytope having cells with q cells joining around an edge is denoted by, the vertex figure of the 4-polytope is a. A five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to
42.
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