1.
5-cell
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In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid and it is a 4-simplex, the simplest possible convex regular 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base, the regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol. Pentachoron 4-simplex Pentatope Pentahedroid Pen Hyperpyramid, tetrahedral pyramid The 5-cell is self-dual and its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1, or approximately 75. 52°, the 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. The simplest set of coordinates is, with edge length 2√2, a 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, the purple edges represent the Petrie polygon of the 5-cell. The A4 Coxeter plane projects the 5-cell into a regular pentagon, the four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral pyramid vertex figures, Other uniform 5-polytopes have irregular 5-cell vertex figures, the symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and this compound has symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell, the pentachoron is the simplest of 9 uniform polychora constructed from the Coxeter group. It is in the sequence of regular polychora, the tesseract, 120-cell, of Euclidean 4-space, all of these have a tetrahedral vertex figure. It is similar to three regular polychora, the tesseract, 600-cell of Euclidean 4-space, and the order-6 tetrahedral honeycomb of hyperbolic space, all of these have a tetrahedral cell. T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
2.
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
3.
Uniform 4-polytope
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In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. 47 non-prismatic convex uniform 4-polytopes, one set of convex prismatic forms. There are also a number of non-convex star forms. Regular star 4-polytopes 1852, Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures and this construction enumerated 45 semiregular 4-polytopes. 1912, E. L. Elte independently expanded on Gossets list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets, Convex uniform polytopes,1940, The search was expanded systematically by H. S. M. Coxeter in his publication Regular and Semi-Regular Polytopes,1966 Norman Johnson completes his Ph. D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher. 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, 1998-2000, The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevskys online indexed enumeration. Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly,2004, A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnsons naming system in his listing,2008, The Symmetries of Things was published by John H. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, nonregular uniform star 4-polytopes, 2000-2005, In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes had been identified by Jonathan Bowers and George Olshevsky. Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements, Regular 4-polytopes can be expressed with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures. The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes cells, there are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms. 5 are polyhedral prisms based on the Platonic solids 13 are polyhedral prisms based on the Archimedean solids 9 are in the self-dual regular A4 group family,9 are in the self-dual regular F4 group family. 15 are in the regular B4 group family 15 are in the regular H4 group family,1 special snub form in the group family. 1 special non-Wythoffian 4-polytopes, the grand antiprism, TOTAL,68 −4 =64 These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets, in addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms, Set of uniform antiprismatic prisms - sr× - Polyhedral prisms of two antiprisms. Set of uniform duoprisms - × - A product of two polygons, the 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way. Facets are given, grouped in their Coxeter diagram locations by removing specified nodes, there is one small index subgroup +, order 60, or its doubling +, order 120, defining a omnisnub 5-cell which is listed for completeness, but is not uniform
4.
Polytope
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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli, the German term polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott. The term polytope is nowadays a broad term that covers a class of objects. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes and they represent different approaches to generalizing the convex polytopes to include other objects with similar properties. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold, an example of this approach defines a polytope as a set of points that admits a simplicial decomposition. However this definition does not allow star polytopes with interior structures, the discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra and this approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms polytope and polyhedron are used in a different sense and this terminology is typically confined to polytopes and polyhedra that are convex. A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells, terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an -dimensional element while others use face to denote a 2-face specifically, authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element, the terms adopted in this article are given in the table below, An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope, Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope and these bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point, a 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, the convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite, Polytopes are defined in this way, e. g. in linear programming
5.
Truncation (geometry)
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In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Keplers names for the Archimedean solids, in general any polyhedron can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, there are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation, for example, the icosidodecahedron, represented as Schläfli symbols r or, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented as tr or t. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the adjacent to the ringed node. A truncated n-sided polygon will have 2n sides, a regular polygon uniformly truncated will become another regular polygon, t is. A complete truncation, r, is another regular polygon in its dual position, a regular polygon can also be represented by its Coxeter-Dynkin diagram, and its uniform truncation, and its complete truncation. Star polygons can also be truncated, a truncated pentagram will look like a pentagon, but is actually a double-covered decagon with two sets of overlapping vertices and edges. A truncated great heptagram gives a tetradecagram and this sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron, the middle image is the uniform truncated cube. It is represented by a Schläfli symbol t, a bitruncation is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. The truncated octahedron is a cube, 2t is an example. A complete bitruncation is called a birectification that reduces original faces to points, for polyhedra, this becomes the dual polyhedron. An octahedron is a birectification of the cube, = 2r is an example, another type of truncation is called cantellation, cuts edge and vertices, removing original edges and replacing them with rectangles. Higher dimensional polytopes have higher truncations, runcination cuts faces, edges, in 5-dimensions sterication cuts cells, faces, and edges. Edge-truncation is a beveling or chamfer for polyhedra, similar to cantellation but retains original vertices, in 4-polytopes edge-truncation replaces edges with elongated bipyramid cells. Alternation or partial truncation only removes some of the original vertices, a partial truncation or alternation - Half of the vertices and connecting edges are completely removed. The operation only applies to polytopes with even-sided faces, faces are reduced to half as many sides, and square faces degenerate into edges
6.
Bitruncation
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In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification, the original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation t1,2 or 2t, for regular polyhedra, a bitruncated form is the truncated dual. For example, a cube is a truncated octahedron. For regular 4-polytope, a form is a dual-symmetric operator. A bitruncated 4-polytope is the same as the bitruncated dual, a regular polytope will have its cells bitruncated into truncated cells, and the vertices are replaced by truncated cells. An interesting result of operation is that self-dual 4-polytope remain cell-transitive after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra, two are honeycombs on the 3-sphere, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space. Uniform polyhedron uniform 4-polytope Rectification Truncation Coxeter, H. S. M, regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Truncation. Archived from the original on 4 February 2007
7.
Schlegel diagram
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In geometry, a Schlegel diagram is a projection of a polytope from R d into R d −1 through a point beyond one of its facets or faces. The resulting entity is a subdivision of the facet in R d −1 that is combinatorially equivalent to the original polytope. Named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes, in dimensions 3 and 4, a Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a 4-polytope to 3-space, respectively. As such, Schlegel diagrams are used as a means of visualizing four-dimensional polytopes. The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows, if it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces and this is called a Schlegel diagram of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron, Sommerville also considers the case of a simplex in four dimensions, The Schlegel diagram of simplex in S4 is a tetrahedron divided into four tetrahedra. More generally, a polytope in n-dimensions has a Schegel diagram constructed by a perspective projection viewed from a point outside of the polytope, all vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection. Net – A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets and this maintains the geometric scale and shape, but makes the topological connections harder to see. Victor Schlegel Theorie der homogen zusammengesetzten Raumgebilde, Nova Acta, Ksl, deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden, Victor Schlegel Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Grünbaum, Branko, Kaibel, Volker, Klee, Victor, convex polytopes, New York & London, Springer-Verlag, ISBN 0-387-00424-6. George W. Hart, 4D Polytope Projection Models by 3D Printing Nrich maths – for the teenager
8.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
9.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950
10.
Truncated tetrahedron
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In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces,4 equilateral triangle faces,12 vertices and 18 edges and it can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron, a truncated tetrahedron is the Goldberg polyhedron GIII, containing triangular and hexagonal faces. A truncated tetrahedron can be called a cube, with Coxeter diagram. There are two positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra. The area A and the volume V of a tetrahedron of edge length a are. The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, in fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. A lower symmetry version of the tetrahedron is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges, Friauf and his 1927 paper The crystal structure of the intermetallic compound MgCu2. Giant truncated tetrahedra were used for the Man the Explorer and Man the Producer theme pavilions in Expo 67 and they were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms, all of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada, the Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations, in the mathematical field of graph theory, a truncated tetrahedral graph is a Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges and it is a connected cubic graph, and connected cubic transitive graph. It is also a part of a sequence of cantic polyhedra, in this wythoff construction the edges between the hexagons represent degenerate digons
11.
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
12.
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
13.
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
14.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
15.
Rectified 5-cell
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In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra, each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces,30 edges, and 10 vertices, each vertex is surrounded by 3 octahedra and 2 tetrahedra, the vertex figure is a triangular prism. The vertex figure of the rectified 5-cell is a triangular prism. Together with the simplex and 24-cell, this shape and its dual was one of the first 2-simple 2-simplicial 4-polytopes known and this means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another pair of examples. The birectified 5-cell can be seen as the intersection of two regular 5-cells in dual positions and it is one of three semiregular 4-polytope made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5. These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively and this polytope is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope. It is also one of 9 Uniform 4-polytopes constructed from the Coxeter group, the rectified 5-cell is second in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed as the figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, the Coxeter symbol for the rectified 5-cell is 021. Semiregular k 21 polytope T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Rectified 5-cell - data, convex uniform polychora based on the pentachoron - Model 2, George Olshevsky
16.
Cantellated 5-cell
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In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation of the regular 5-cell. There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations, the cantellated 5-cell or small rhombated pentachroron is a uniform 4-polytope. It has 30 vertices,90 edges,80 faces, and 20 cells, the cells are 5 cuboctahedra,5 octahedra, and 10 triangular prisms. Each vertex is surrounded by 2 cuboctahedra,2 triangular prisms, and 1 octahedron, the cantitruncated 5-cell or great rhombated pentachoron is a uniform 4-polytope. It is composed of 60 vertices,120 edges,80 faces, the cells are,5 truncated octahedra,10 triangular prisms, and 5 truncated tetrahedra. Each vertex is surrounded by 2 truncated octahedra, one triangular prism and these polytopes are art of a set of 9 Uniform 4-polytopes constructed from the Coxeter group. 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.1, convex uniform polychora based on the pentachoron - Model 4,7, George Olshevsky. X3o3x3o - srip, x3x3x3o - grip
17.
Cell (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
18.
F-vector
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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
19.
Wythoff construction
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In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoffs kaleidoscopic construction and it is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope, however, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point, if the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a point inside the spherical triangle enclosed by the mirrors. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron and this produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and similarly for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C and this produces a polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC and this produces a polyhedron with Wythoff symbol a b c|. The vertex is at a point such that, when it is rotated around any of the corners by twice the angle at that point. Only even-numbered reflections of the vertex are used. The polyhedron has the Wythoff symbol |a b c, the process in general also applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes. Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian and they generally can be derived from Wythoffian forms either by alternation or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry, sometimes snub forms are considered Wythoffian, even though they can only be constructed by the alternation of omnitruncated forms. Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra, coxeter-Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs. Coxeter Regular Polytopes, Third edition, Dover edition, ISBN 0-486-61480-8 Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 HarEl, Z. W. A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences,20 966–970, archived from the original on 4 February 2007. Displays Uniform Polyhedra using Wythoffs construction method Description of Wythoff Constructions Jenn, software that generates views of polyhedra and polychora from symmetry groups
20.
K-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
21.
Triangular pyramid
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
22.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
23.
Isosceles triangle
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
24.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
25.
Coxeter plane
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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
26.
Dihedral symmetry
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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
27.
Net (polytope)
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In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are an aid to the study of polyhedra and solid geometry in general. Many different nets can exist for a polyhedron, depending on the choices of which edges are joined. Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded, additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra. Shephard asked whether every convex polyhedron has at least one net and this question, which is also known as Dürers conjecture, or Dürers unfolding problem, remains unanswered. There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron so that the set of subdivided faces has a net, in 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an affine transformation. The shortest path over the surface between two points on the surface of a polyhedron corresponds to a line on a suitable net for the subset of faces touched by the path. The net has to be such that the line is fully within it. Other candidates for the shortest path are through the surface of a third face adjacent to both, and corresponding nets can be used to find the shortest path in each category, the geometric concept of a net can be extended to higher dimensions. The above net of the tesseract, the hypercube, is used prominently in a painting by Salvador Dalí. However, it is known to be possible for every convex uniform 4-polytope, Paper model Cardboard modeling UV mapping Weisstein, Eric W. Net. Regular 4d Polytope Foldouts Editable Printable Polyhedral Nets with an Interactive 3D View Paper Models of Polyhedra Unfolder for Blender Unfolding package for Mathematica
28.
Stereographic projection
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In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =
29.
Simplex
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In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices, a regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex, the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨
30.
Cartesian coordinate
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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
31.
Orthant
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In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces, by independent selections of half-space signs, there are 2n orthants in n-dimensional space. More specifically, an orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities, ε1x1 ≥0 ε2x2 ≥0 · · · εnxn ≥0, where each εi is +1 or −1. Similarly, an orthant in Rn is a subset defined by a system of strict inequalities ε1x1 >0 ε2x2 >0 · · · εnxn >0. By dimension, In one dimension, an orthant is a ray, in two dimensions, an orthant is a quadrant. In three dimensions, an orthant is an octant, john Conway defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2n simplex facets, one per orthant. Cross polytope - a family of polytopes in n-dimensions which can be constructed with one simplex facets in each orthant space. Measure polytope - a family of polytopes in n-dimensions which can be constructed with one vertex in each orthant space. Orthotope - Generalization of a rectangle in n-dimensions, with one vertex in each orthant, the facts on file, Geometry handbook, Catherine A. Gorini,2003, ISBN 0-8160-4875-4, p.113
32.
Truncated pentacross
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In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex, vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second, all edges are shortened, and two new vertices are added on each original edge. The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb, all edges are shortened, and two new vertices are added on each original edge. This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex, coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. X3x3o3o4o - tot, x3x3x3o4o - gart Weisstein, Eric W. Hypercube, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
33.
Bitruncated penteract
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In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube. There are four unique truncations of the 5-cube, vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the faces of the 5-cube. The third and fourth truncations are more easily constructed as second, truncated penteract The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at 1 / of the edge length. A regular 5-cell is formed at each truncated vertex, the Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of, The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge, coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. O3o3o3x4x - tan, o3o3x3x4o - bittin Weisstein, Eric W. Hypercube, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
34.
Tetragonal disphenoid
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In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which two edges that are opposite each other have equal lengths. Other names for the shape are sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron. All the solid angles and vertex figures of a disphenoid are the same, however, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths. If the faces of a disphenoid are equilateral triangles, it is a tetrahedron with Td tetrahedral symmetry. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid, in this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid, unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. Both tetragonal disphenoids and rhombic disphenoids are isohedra, as well as being congruent to each other and it is not possible to construct a disphenoid with right triangle or obtuse triangle faces. When right triangles are glued together in the pattern of a disphenoid, two more types of tetrahedron generalize the disphenoid and have similar names. The digonal disphenoid has faces with two different shapes, both triangles, with two faces of each shape. The phyllic disphenoid similarly has faces with two shapes of scalene triangles, disphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms. A tetrahedron is a if and only if its circumscribed parallelepiped is right-angled. We also have that a tetrahedron is a if and only if the center in the circumscribed sphere. The disphenoids are the polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting and they are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints. The volume of a disphenoid with opposite edges of length l, m and n is given by V =72. There is also the following interesting relation connecting the volume and the circumradius,16 T2 R2 = l 2 m 2 n 2 +9 V2, the squares of the lengths of the bimedians are 12,12,12. If the four faces of a tetrahedron have the same perimeter, if the four faces of a tetrahedron have the same area, then it is a disphenoid
35.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
36.
Isochoric figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
37.
Runcinated pentachoron
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In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including permutations, truncations. The runcinated 5-cell or small prismatodecachoron is constructed by expanding the cells of a 5-cell radially and filling in the gaps with triangular prisms and it consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual, E. L. Elte identified it in 1912 as a semiregular polytope. Runcinated 5-cell Runcinated pentachoron Runcinated 4-simplex Expanded 5-cell/4-simplex/pentachoron Small prismatodecachoron Two of the ten tetrahedral cells meet at each vertex, the triangular prisms lie between them, joined to them by their triangular faces and to each other by their square faces. Each triangular prism is joined to its neighbouring triangular prisms in anti orientation, thus each pair of adjacent prisms, if rotated into the same hyperplane, the runcinated 5-cell can be dissected by a central cuboctahedron into two tetrahedral cupola. This dissection is analogous to the 3D cuboctahedron being dissected by a hexagon into two triangular cupola. A second construction in 5-space, from the center of a rectified 5-orthoplex is given by coordinate permutations of and it is also the vertex figure for the 5-cell honeycomb in 4-space. The maximal cross-section of the runcinated 5-cell with a 3-dimensional hyperplane is a cuboctahedron and this cross-section divides the runcinated 5-cell into two tetrahedral hypercupolae consisting of 5 tetrahedra and 10 triangular prisms each. The tetrahedron-first orthographic projection of the runcinated 5-cell into 3-dimensional space has a cuboctahedral envelope and these are the images of 5 of the tetrahedral cells. The 6 square faces of the cuboctahedron are joined to the edges of the central tetrahedron via distorted triangular prisms and these are the images of 6 of the triangular prism cells. The other 4 triangular faces are joined to the central tetrahedron via 4 triangular prisms and these are the images of another 4 of the triangular prism cells. This accounts for half of the runcinated 5-cell, which may be thought of as the northern hemisphere. The other half, the hemisphere, corresponds to an isomorphic division of the cuboctahedron in dual orientation. The triangular faces of the join the triangular prisms in one hemisphere to the flattened tetrahedra in the other hemisphere. Thus, the southern hemisphere contains another 5 tetrahedra and another 10 triangular prisms, the regular skew polyhedron, exists in 4-space with 6 squares around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 5-cell, using all 60 edges and 20 vertices, the 40 triangular faces of the runcinated 5-cell can be seen as removed. The dual regular skew polyhedron, is related to the hexagonal faces of the bitruncated 5-cell
38.
Cantitruncated 5-cell
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In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation of the regular 5-cell. There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations, the cantellated 5-cell or small rhombated pentachroron is a uniform 4-polytope. It has 30 vertices,90 edges,80 faces, and 20 cells, the cells are 5 cuboctahedra,5 octahedra, and 10 triangular prisms. Each vertex is surrounded by 2 cuboctahedra,2 triangular prisms, and 1 octahedron, the cantitruncated 5-cell or great rhombated pentachoron is a uniform 4-polytope. It is composed of 60 vertices,120 edges,80 faces, the cells are,5 truncated octahedra,10 triangular prisms, and 5 truncated tetrahedra. Each vertex is surrounded by 2 truncated octahedra, one triangular prism and these polytopes are art of a set of 9 Uniform 4-polytopes constructed from the Coxeter group. 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.1, convex uniform polychora based on the pentachoron - Model 4,7, George Olshevsky. X3o3x3o - srip, x3x3x3o - grip
39.
Bitruncation (geometry)
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In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification, the original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation t1,2 or 2t, for regular polyhedra, a bitruncated form is the truncated dual. For example, a cube is a truncated octahedron. For regular 4-polytope, a form is a dual-symmetric operator. A bitruncated 4-polytope is the same as the bitruncated dual, a regular polytope will have its cells bitruncated into truncated cells, and the vertices are replaced by truncated cells. An interesting result of operation is that self-dual 4-polytope remain cell-transitive after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra, two are honeycombs on the 3-sphere, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space. Uniform polyhedron uniform 4-polytope Rectification Truncation Coxeter, H. S. M, regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Truncation. Archived from the original on 4 February 2007
40.
4-polytope
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In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements, vertices, edges, faces, each face is shared by exactly two cells. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron, topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space, similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space, a 4-polytope is a closed four-dimensional figure. It comprises vertices, edges, faces and cells, a cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i. e. it is not a compound, the most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube. 4-polytopes cannot be seen in space due to their extra dimension. Several techniques are used to help visualise them, Orthogonal projection Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes. Perspective projection Just as a 3D shape can be projected onto a flat sheet, sectioning Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut hypersurface in three dimensions. A sequence of sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce an animation of these cross sections. The topology of any given 4-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, like all polytopes, 4-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the shapes of the non-convex star polygons. A 4-polytope is regular if it is transitive on its flags and this means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron
41.
Cell (mathematics)
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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
42.
Intersection (set theory)
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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers. The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can also be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
43.
Pentachoron
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In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid and it is a 4-simplex, the simplest possible convex regular 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base, the regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol. Pentachoron 4-simplex Pentatope Pentahedroid Pen Hyperpyramid, tetrahedral pyramid The 5-cell is self-dual and its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1, or approximately 75. 52°, the 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. The simplest set of coordinates is, with edge length 2√2, a 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, the purple edges represent the Petrie polygon of the 5-cell. The A4 Coxeter plane projects the 5-cell into a regular pentagon, the four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral pyramid vertex figures, Other uniform 5-polytopes have irregular 5-cell vertex figures, the symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and this compound has symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell, the pentachoron is the simplest of 9 uniform polychora constructed from the Coxeter group. It is in the sequence of regular polychora, the tesseract, 120-cell, of Euclidean 4-space, all of these have a tetrahedral vertex figure. It is similar to three regular polychora, the tesseract, 600-cell of Euclidean 4-space, and the order-6 tetrahedral honeycomb of hyperbolic space, all of these have a tetrahedral cell. T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
44.
Penteract
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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
45.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron 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. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size