1.
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
2.
120-cell
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In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C120, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid, the boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the dodecahedron and has called a dodecaplex. Just as a dodecahedron can be built up as a model with 12 pentagons,3 around each vertex, there are 120 cells,720 pentagonal faces,1200 edges, and 600 vertices. There are 4 dodecahedra,6 pentagons, and 4 edges meeting at every vertex, there are 3 dodecahedra and 3 pentagons meeting every edge. The dual polytope of the 120-cell is the 600-cell, the vertex figure of the 120-cell is a tetrahedron. The dihedral angle of the 120-cell is 144° The 600 vertices of the 120-cell include all permutations of, the 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces, one can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use, a stereographic projection, and a structure of intertwining rings. The cell locations lend themselves to a hyperspherical description, pick an arbitrary cell and label it the North Pole. Twelve great circle meridians radiate out in 3 dimensions, converging at the 5th South Pole cell and this skeleton accounts for 50 of the 120 cells. Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, the cells labeled interstitial in the following table do not fall on meridian great circles. Layers 2,4,6 and 8 cells are located over the cells faces. Layers 3 and 7s cells are located directly over the pole cells vertices, layer 5s cells are located over the pole cells edges. The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring, although the outer rings spiral around the inner ring, they actually have no helical torsion. The spiraling is a result of the 3-sphere curvature, the inner ring and the five outer rings now form a six ring, 60-cell solid torus
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.
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
5.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
6.
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
7.
Pentagonal prism
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In geometry, the pentagonal prism a prism with a pentagonal base. It is a type of heptahedron with 7 faces,15 edges and it can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a pentagon and a line segment. The dual of a prism is a pentagonal bipyramid. The symmetry group of a pentagonal prism is D5h of order 20. The rotation group is D5 of order 10, the volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. It exists as cells of four nonprismatic uniform 4-polytopes in 4 dimensions, Weisstein, Pentagonal Prism Polyhedron Model -- works in your web browser
8.
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
9.
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
10.
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
11.
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
12.
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
13.
Rectification (geometry)
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In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the facets of the original polytope. A rectification operator is denoted by the symbol r, for example, r is the rectified cube. Conway polyhedron notation uses ambo for this operator, in graph theory this operation creates a medial graph. Rectification is the point of a truncation process. The highest degree of rectification creates the dual polytope, a rectification truncates edges to points. A birectification truncates faces to points, a trirectification truncates cells to points, and so on. New vertices are placed at the center of the edges of the original polygon, each platonic solid and its dual have the same rectified polyhedron. The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual, the rectified octahedron, whose dual is the cube, is the cuboctahedron. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron, a rectified square tiling is a square tiling. A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling, examples If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. The resulting medial graph remains polyhedral, so by Steinitzs theorem it can be represented as a polyhedron, the Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, is Conways expand operation, e, which is the same as Johnsons cantellation operation, t0,2 generated from regular polyhedral, each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. Its rectification will have two types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex. A rectified is not the same as a rectified, however, a further truncation, called bitruncation, is symmetric between a 4-polytope and its dual. Examples A first rectification truncates edges down to points, If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1 or r. A second rectification, or birectification, truncates faces down to points, If regular it has notation t2 or 2r. For polyhedra, a birectification creates a dual polyhedron, higher degree rectifications can be constructed for higher dimensional polytopes
14.
600-cell
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In geometry, the 600-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C600, hexacosichoron and hexacosidedroid, the 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex and polytetrahedron, being bounded by tetrahedral cells and its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces,720 edges, and 120 vertices, the edges form 72 flat regular decagons. Each vertex of the 600-cell is a vertex of six such decagons, References, S. L. van Oss, F. Buekenhout and M. Parker. Its vertex figure is an icosahedron, and its dual polytope is the 120-cell and it has a dihedral angle of cos−1 = ~164. 48°. Each cell touches, in manner,56 other cells. One cell contacts each of the four faces, two cells contact each of the six edges, but not a face, and ten cells contact each of the four vertices, but not a face or edge. The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ, can be given as follows,16 vertices of the form, the remaining 96 vertices are obtained by taking even permutations of ½. Note that the first 16 vertices are the vertices of a tesseract, the eight are the vertices of a 16-cell. The final 96 vertices are the vertices of a snub 24-cell, when interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. Each rotational symmetry of the 600-cell is generated by elements of 2IL and 2IR. The centre of RSG consists of the non-rotation Id and the central inversion -Id and we have the isomorphism RSG ≅ /. The order of RSG equals 120 ×120 /2 =7200, the binary icosahedral group is isomorphic to SL. The full symmetry group of the 600-cell is the Weyl group of H4 and this is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections, the rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S. L. van Oss, one can start by realizing the 600-cell is the dual of the 120-cell
15.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
16.
Icosidodecahedron
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In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly and its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, the icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae. In this form its symmetry is D5d, order 20, the wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the permutations of. The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a face. The last two correspond to the A2 and H2 Coxeter planes, the icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The icosidodecahedron is a dodecahedron and also a rectified icosahedron. With orbifold notation symmetry of all of these tilings are wythoff construction within a fundamental domain of symmetry. The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images, the icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves. Eight uniform star polyhedra share the same vertex arrangement, of these, two also share the same edge arrangement, the small icosihemidodecahedron, and the small dodecahemidodecahedron. The vertex arrangement is shared with the compounds of five octahedra. In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words, the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons, six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron, in the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids
17.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
18.
Semiregular polytope
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In geometry, by Thorold Gossets definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition, in three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell, the only semiregular polytopes in higher dimensions are the k21 polytopes, where the rectified 5-cell is the special case of k =0. These were all listed by Gosset, but a proof of the completeness of this list was not published until the work of Makarov for four dimensions, the semiregular Euclidean honeycombs are the tetrahedral-octahedral honeycomb, gyrated alternated cubic honeycomb and the 521 honeycomb. Coxeter, H. S. M. Regular Polytopes, coxeter, H. S. M. Whitrow, G. J. World-structure and non-Euclidean honeycombs. The Semiregular Polytopes of the Hyperspaces, on the regular and semi-regular figures in space of n dimensions. On the derivation of four-dimensional semi-regular polytopes
19.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
20.
Coxeter element
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element is a product of all simple reflections, the product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections
21.
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 =
22.
Net (polyhedron)
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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
23.
Square pyramid
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In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, if the sides are all equilateral triangles, the pyramid is one of the Johnson solids. The 92 Johnson solids were named and described by Norman Johnson in 1966, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, the Johnson square pyramid can be characterized by a single edge-length parameter a. The height H, the surface area A, and the volume V of such a pyramid are, other square pyramids have isosceles triangle sides. For square pyramids in general, with length l and height h. Square pyramids fill space with tetrahedra, truncated cubes or cuboctahedra, the square pyramid is topologically a self-dual polyhedron. The dual edge lengths are different due to the polar reciprocation, like all pyramids, the square pyramid is self-dual, having the same number of vertices as faces. A square pyramid can be represented by the Wheel graph W5, eric W. Weisstein, Square pyramid at MathWorld. Square Pyramid -- Interactive Polyhedron Model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra
24.
Pentagonal antiprism
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In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two joined to each other by a ring of 10 triangles for a total of 12 faces. Hence, it is a non-regular dodecahedron, if the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron. The two pentagonal faces of either shape can be augmented with pyramids to form the icosahedron, the pentagonal antiprism occurs as a constituent element in some higher-dimensional polytopes. Two rings of 10 pentagonal antiprisms each bound the hypersurface of the 4-dimensional grand antiprism, if these antiprisms are augmented with pentagonal prism pyramids and linked with rings of 5 tetrahedra each, the 600-cell is obtained. The pentagonal antiprism can be truncated and alternated to form a snub antiprism, Weisstein, pentagonal Antiprism, Interactive Polyhedron Model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A5
25.
Rectified 120-cell
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In geometry, a rectified 120-cell is a uniform 4-polytope formed as the rectification of the regular 120-cell. E. L. Elte identified it in 1912 as a semiregular polytope, there are four rectifications of the 120-cell, including the zeroth, the 120-cell itself. The birectified 120-cell is more seen as a rectified 600-cell. In geometry, the rectified 120-cell or rectified hecatonicosachoron is a convex uniform 4-polytope composed of 600 regular tetrahedra and 120 icosidodecahedra cells and its vertex figure is a triangular prism, with three icosidodecahedra and two tetrahedra meeting at each vertex. 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, J. H. Conway, guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39,1965 N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 Convex uniform polychora based on the hecatonicosachoron and hexacosichoron - Model 33, George Olshevsky. Rectified 120-cell Marco Möllers Archimedean polytopes in R4 Klitzing, Richard, 4D uniform polytopes o3o3x5o - rahi. Four-dimensional Archimedean Polytopes, Marco Möller,2004 PhD dissertation H4 uniform polytopes with coordinates, r
26.
Truncated 120-cells
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In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell. There are three trunctions, including a bitruncation, and a tritruncation, which creates the truncated 600-cell, the truncated 120-cell or truncated hecatonicosachoron is a uniform 4-polytope, constructed by a uniform truncation of the regular 120-cell 4-polytope. It is made of 120 truncated dodecahedral and 600 tetrahedral cells and it has 3120 faces,2400 being triangles and 720 being decagons. There are 4800 edges of two types,3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron, each vertex has 3 truncated dodecahedra and one tetrahedron around it. Its vertex figure is a triangular pyramid. It has 720 cells,120 truncated icosahedra, and 600 truncated tetrahedra and its vertex figure is a digonal disphenoid, with two truncated icosahedra and two truncated tetrahedra around it. It is derived from the 600-cell by truncation and it has 720 cells,120 icosahedra and 600 truncated tetrahedra. Its vertex figure is a pyramid, with one icosahedron on the base. Truncated 600-cell Truncated hexacosichoron Truncated tetraplex The truncated 600-cell consists of 600 truncated tetrahedra and 120 icosahedra, the truncated tetrahedral cells are joined to each other via their hexagonal faces, and to the icosahedral cells via their triangular faces. Each icosahedron is surrounded by 20 truncated tetrahedra, 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, J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. O3o3x5x - thi, o3x3x5o - xhi, x3x3o5o - tex Four-Dimensional Polytope Projection Barn Raisings, George W. Hart H4 uniform polytopes with coordinates, t t 2t
27.
Cantellated 120-cell
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In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation of the regular 120-cell. There are four degrees of cantellations of the 120-cell including with permutations truncations, two are expressed relative to the dual 600-cell. The cantellated 120-cell is a uniform 4-polytope and it is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra,1200 triangular prisms,600 octahedra and its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex. This 4-polytope is related to the regular 120-cell, the cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells, the image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the faces are hidden, allowing us to see the elemented projected inside. It has 1440 cells,120 icosidodecahedra,600 cuboctahedra, and 720 pentagonal prisms and its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms, there are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra. There are two classes of edges, 3-4-4, 3-4-5,3600 have two squares and a triangle around it, and 7200 have one triangle, one square, the cantitruncated 600-cell is a uniform 4-polytope. It is composed of 1440 cells,120 truncated icosahedra,720 pentagonal prisms and 600 truncated octahedra and it has 7200 vertices,14400 edges, and 8640 faces. It has a tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism. Archimedisches Polychor Nr.57 Marco Möllers Archimedean polytopes in R4 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, J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. H4 uniform polytopes with coordinates, rr rr tr tr
28.
Runcinated 120-cells
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In four-dimensional geometry, a runcinated 120-cell is a convex uniform 4-polytope, being a runcination of the regular 120-cell. There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations, the runcinated 120-cell can be seen as an expansion applied to a regular 4-polytope, the 120-cell or 600-cell. The runcinated 120-cell or small disprismatohexacosihecatonicosachoron is a uniform 4-polytope and it has 2640 cells,120 dodecahedra,720 pentagonal prisms,1200 triangular prisms, and 600 tetrahedra. Its vertex figure is a triangular antiprism, its bases represent a dodecahedron and a tetrahedron. It contains 2640 cells,120 truncated dodecahedra,720 decagonal prisms,1200 triangular prisms and its vertex figure is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron. Runcicantellated 600-cell Prismatorhombated hexacosichoron The runcitruncated 600-cell or prismatorhombated hecatonicosachoron is a uniform 4-polytope and it is composed of 2640 cells,120 rhombicosidodecahedron,600 truncated tetrahedra,720 pentagonal prisms, and 1200 hexagonal prisms. It has 7200 vertices,18000 edges, and 13440 faces and it has 14400 vertices,28800 edges, and 17040 faces. It is the largest nonprismatic convex uniform 4-polytope, the vertices and edges form the Cayley graph of the Coxeter group H4. It has 9840 cells,35040 faces,32400 edges, and 7200 vertices and these polytopes are a part of a set of 15 uniform 4-polytopes with H4 symmetry, 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, J. H. Conway, guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39,1965 N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3o3o5x - sidpixhi, x3o3x5x - prix, x3x3o5x - prahi, x3x3x5x - gidpixhi H4 uniform polytopes with coordinates, t03 t013 t013 t0123
29.
Cantitruncated 120-cell
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In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation of the regular 120-cell. There are four degrees of cantellations of the 120-cell including with permutations truncations, two are expressed relative to the dual 600-cell. The cantellated 120-cell is a uniform 4-polytope and it is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra,1200 triangular prisms,600 octahedra and its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex. This 4-polytope is related to the regular 120-cell, the cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells, the image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the faces are hidden, allowing us to see the elemented projected inside. It has 1440 cells,120 icosidodecahedra,600 cuboctahedra, and 720 pentagonal prisms and its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms, there are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra. There are two classes of edges, 3-4-4, 3-4-5,3600 have two squares and a triangle around it, and 7200 have one triangle, one square, the cantitruncated 600-cell is a uniform 4-polytope. It is composed of 1440 cells,120 truncated icosahedra,720 pentagonal prisms and 600 truncated octahedra and it has 7200 vertices,14400 edges, and 8640 faces. It has a tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism. Archimedisches Polychor Nr.57 Marco Möllers Archimedean polytopes in R4 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, J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. H4 uniform polytopes with coordinates, rr rr tr tr
30.
Cantellated 600-cell
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In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation of the regular 120-cell. There are four degrees of cantellations of the 120-cell including with permutations truncations, two are expressed relative to the dual 600-cell. The cantellated 120-cell is a uniform 4-polytope and it is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra,1200 triangular prisms,600 octahedra and its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex. This 4-polytope is related to the regular 120-cell, the cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells, the image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the faces are hidden, allowing us to see the elemented projected inside. It has 1440 cells,120 icosidodecahedra,600 cuboctahedra, and 720 pentagonal prisms and its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms, there are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra. There are two classes of edges, 3-4-4, 3-4-5,3600 have two squares and a triangle around it, and 7200 have one triangle, one square, the cantitruncated 600-cell is a uniform 4-polytope. It is composed of 1440 cells,120 truncated icosahedra,720 pentagonal prisms and 600 truncated octahedra and it has 7200 vertices,14400 edges, and 8640 faces. It has a tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism. Archimedisches Polychor Nr.57 Marco Möllers Archimedean polytopes in R4 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, J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. H4 uniform polytopes with coordinates, rr rr tr tr
31.
Cantitruncated 600-cell
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In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation of the regular 120-cell. There are four degrees of cantellations of the 120-cell including with permutations truncations, two are expressed relative to the dual 600-cell. The cantellated 120-cell is a uniform 4-polytope and it is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra,1200 triangular prisms,600 octahedra and its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex. This 4-polytope is related to the regular 120-cell, the cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells, the image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the faces are hidden, allowing us to see the elemented projected inside. It has 1440 cells,120 icosidodecahedra,600 cuboctahedra, and 720 pentagonal prisms and its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms, there are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra. There are two classes of edges, 3-4-4, 3-4-5,3600 have two squares and a triangle around it, and 7200 have one triangle, one square, the cantitruncated 600-cell is a uniform 4-polytope. It is composed of 1440 cells,120 truncated icosahedra,720 pentagonal prisms and 600 truncated octahedra and it has 7200 vertices,14400 edges, and 8640 faces. It has a tetrahedral vertex figure, filled by one truncated icosahedron, one pentagonal prism. Archimedisches Polychor Nr.57 Marco Möllers Archimedean polytopes in R4 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, J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. H4 uniform polytopes with coordinates, rr rr tr tr