1.
Prismatic uniform polyhedron
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In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids, because they are isogonal, their vertex arrangement uniquely corresponds to a symmetry group. Each has p reflection planes which contain the p-fold axis, the Dph symmetry group contains inversion if and only if p is even, while Dpd contains inversion symmetry if and only if p is odd. There are, prisms, for each rational number p/q >2, with symmetry group Dph, antiprisms, for each rational number p/q > 3/2, with symmetry group Dpd if q is odd, Dph if q is even. If p/q is an integer, i. e. if q =1, an antiprism with p/q <2 is crossed or retrograde, its vertex figure resembles a bowtie. If p/q ≤ 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality, Uniform polyhedron Prism Antiprism Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Philosophical Transactions of the Royal Society of London, P.175 Skilling, John, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society,79, 447–457, doi,10. 1017/S0305004100052440, MR0397554. Prisms and Antiprisms George W. Hart
2.
Euler characteristic
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It is commonly denoted by χ. The Euler characteristic was originally defined for polyhedra and used to prove theorems about them. Leonhard Euler, for whom the concept is named, was responsible for much of early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, any convex polyhedrons surface has Euler characteristic V − E + F =2. This equation is known as Eulers polyhedron formula and it corresponds to the Euler characteristic of the sphere, and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below and this version holds both for convex polyhedra and the non-convex Kepler-Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0. The Euler characteristic can be defined for connected plane graphs by the same V − E + F formula as for polyhedral surfaces, the Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of determined by G. For trees, E = V −1 and F =1, if G has C components, the same argument by induction on F shows that V − E + F − C =1. One of the few graph theory papers of Cauchy also proves this result, via stereographic projection the plane maps to the two-dimensional sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchys proof of Eulers formula given below, there are many proofs of Eulers formula. One was given by Cauchy in 1811, as follows and it applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface, after this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, therefore, proving Eulers formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that arent connected yet. This adds one edge and one face and does not change the number of vertices, continue adding edges in this manner until all of the faces are triangular. This decreases the number of edges and faces by one each and does not change the number of vertices, remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph
3.
Wythoff symbol
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In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra, a Wythoff symbol consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators, with a slight extension, Wythoffs symbol can be applied to all uniform polyhedra. However, the methods do not lead to all uniform tilings in euclidean or hyperbolic space. In three dimensions, Wythoffs construction begins by choosing a point on the triangle. If the distance of this point from each of the sides is non-zero, a perpendicular line is then dropped between the generator point and every face that it does not lie on. The three numbers in Wythoffs symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, the triangle is also represented with the same numbers, written. In this notation the mirrors are labeled by the reflection-order of the opposite vertex, the p, q, r values are listed before the bar if the corresponding mirror is active. The one impossible symbol | p q r implies the point is on all mirrors. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, the resulting figure has rotational symmetry only. The generator point can either be on or off each mirror and this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. A node is circled if the point is not on the mirror. There are seven generator points with each set of p, q, r, | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isnt Wythoff-constructible, There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the many such patterns in the hyperbolic plane are also listed. The list of Schwarz triangles includes rational numbers, and determine the set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a domain, colored by even. Selected tilings created by the Wythoff construction are given below, for a more complete list, including cases where r ≠2, see List of uniform polyhedra by Schwarz triangle
4.
List of spherical symmetry groups
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Spherical symmetry groups are also called point groups in three dimensions, however, this article is limited to the finite symmetries. There are five fundamental symmetry classes which have triangular fundamental domains, dihedral, cyclic, tetrahedral, octahedral and this article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups quaternion algebraic structure, the group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, ±, prefix, which implies a central inversion. The crystallography groups,32 in total, are a subset with element orders 2,3,4 and 6, there are four involutional groups, no symmetry, reflection symmetry, 2-fold rotational symmetry, and central point symmetry. There are four infinite cyclic symmetry families, with n=2 or higher, there are three infinite dihedral symmetry families, with n as 2 or higher. There are three types of symmetry, tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries. Crystallographic point group Triangle group List of planar symmetry groups Point groups in two dimensions Peter R. Cromwell, Polyhedra, Appendix I Sands, Donald E, mineola, New York, Dover Publications, Inc. p.165. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, N. W, johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Finite spherical symmetry groups Weisstein, Eric W. Schoenflies symbol. Weisstein, Eric W. Crystallographic point groups, simplest Canonical Polyhedra of Each Symmetry Type, by David I
5.
Dihedral symmetry in three dimensions
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In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn. There are 3 types of symmetry in three dimensions, each shown below in 3 notation, Schönflies notation, Coxeter notation. For n = ∞ they correspond to three frieze groups, Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal is used with respect to an axis of rotation. In 2D the symmetry group Dn includes reflections in lines, in 3D the two operations are distinguished, the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, with reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh. Dnd, has vertical mirror planes between the rotation axes, not through them. As a result the vertical axis is a 2n-fold rotoreflection axis, Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the group for a regular n-sided antiprism. Dn is the group of a partially rotated prism. D2 +, of order 4 is one of the three symmetry group types with the Klein four-group as abstract group and it has three perpendicular 2-fold rotation axes. It is the group of a cuboid with an S written on two opposite faces, in the same orientation. D2h, of order 8 is the group of a cuboid D2d. For Dnh, order 4n Cnh, order 2n Cnv, order 2n Dn, +, order 2n For Dnd, order 4n S2n, order 2n Cnv, order 2n Dn, +, cS1 maint, Multiple names, authors list N. W. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Conway, John Horton, Huson, Daniel H
6.
Point groups in three dimensions
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In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups, accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more fixed points. We choose the origin as one of them, the rotation group of an object is equal to its full symmetry group if and only if the object is chiral. Finite Coxeter groups are a set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram, Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E+, which consists of direct isometries, i. e. isometries preserving orientation, it contains those that leave the origin fixed. O is the product of SO and the group generated by inversion. An example would be C4 for H and S4 for M, Thus M is obtained from H by inverting the isometries in H ∖ L. This is clarifying when categorizing isometry groups, see below, in 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of rotations about that axis is a normal subgroup of the group of all rotations about that axis. e. See also the similar overview including translations, when comparing the symmetry type of two objects, the origin is chosen for each separately, i. e. they need not have the same center. Moreover, two objects are considered to be of the symmetry type if their symmetry groups are conjugate subgroups of O. The conjugacy definition would allow a mirror image of the structure, but this is not needed. For example, if a symmetry group contains a 3-fold axis of rotation, there are many infinite isometry groups, for example, the cyclic group generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical abelian groups by adding more rotations around the same axis, there are also non-abelian groups generated by rotations around different axes. They will be infinite unless the rotations are specially chosen, all the infinite groups mentioned so far are not closed as topological subgroups of O
7.
Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra, there are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a case of the concept of uniform polytope. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, by a polygon they implicitly mean a polygon in 3-dimensional Euclidean space, these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron, if the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate and these require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows, some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra, some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron, there double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra, there are several polyhedra with doubled faces produced by Wythoffs construction. Most authors do not allow doubled faces and remove them as part of the construction, skillings figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra. Regular convex polyhedra, The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato, Theaetetus, Timaeus of Locri, the Etruscans discovered the regular dodecahedron before 500 BC. Nonregular uniform convex polyhedra, The cuboctahedron was known by Plato, Archimedes discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra, piero della Francesca rediscovered the five truncation of the Platonic solids, truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. Luca Pacioli republished Francescas work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, regular star polyhedra, Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two
8.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
9.
Hexagonal dipyramid
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A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces,8 vertices and 18 edges, the 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces and it is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is associated with the regular polyhedral form with pentagonal faces. The term dodecadeltahedron is sometimes used to distinguish the bipyramid from the Platonic solid, the hexagonal bipyramid has a plane of symmetry where the bases of the two pyramids are joined. This plane is a regular hexagon, there are also six planes of symmetry crossing through the two apices. These planes are rhombic and lie at 30° angles to each other, with an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. Hexagonal trapezohedron A similar 12-sided polyhedron with a twist and kite faces, snub disphenoid Another 12-sided polyhedron with 2-fold symmetry and only triangular faces. Archived from the original on 4 February 2007, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model hexagonal dipyramid Conway Notation for Polyhedra Try, dP6
10.
Convex set
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In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. For example, a cube is a convex set, but anything that is hollow or has an indent, for example. The boundary of a set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the hull of A. It is the smallest convex set containing A, a convex function is a real-valued function defined on an interval with the property that its epigraph is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets, the branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. The notion of a set can be generalized as described below. Let S be a space over the real numbers, or, more generally. A set C in S is said to be if, for all x and y in C and all t in the interval. In other words, every point on the segment connecting x and y is in C. This implies that a set in a real or complex topological vector space is path-connected. Furthermore, C is strictly convex if every point on the segment connecting x and y other than the endpoints is inside the interior of C. A set C is called convex if it is convex. The convex subsets of R are simply the intervals of R, some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids, the Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called a non-convex set, the complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. If S is a set in n-dimensional space, then for any collection of r, r >1. Ur in S, and for any nonnegative numbers λ1, + λr =1, then one has, ∑ k =1 r λ k u k ∈ S
11.
Zonohedron
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A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of segments in three-dimensional space. Zonohedra were originally defined and studied by E. S. Fedorov, more generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron, each primary parallelohedron is combinatorially equivalent to one of five types, the rhombohedron, hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron. Let be a collection of three-dimensional vectors, with each vector vi we may associate a line segment. The Minkowski sum forms a zonohedron, and all zonohedra that contain the origin have this form, the vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, by choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. Generators parallel to the edges of an octahedron form a truncated octahedron, the Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Both of these zonohedra are simple, as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles. Any simple zonohedron corresponds in this way to a simplicial arrangement, simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane, which were studied by Grünbaum. There are also many examples that do not fit into these three families. Any prism over a polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular, two faces are equal to the regular polygon from which the prism was formed. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, the truncated cuboctahedron, with 12 squares,8 hexagons, and 6 octagons. The truncated icosidodecahedron, with 30 squares,20 hexagons and 12 decagons, in addition, certain Catalan solids are again zonohedra, The rhombic dodecahedron is the dual of the cuboctahedron. The rhombic triacontahedron is the dual of the icosidodecahedron, zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron
12.
Vertex configuration
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In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
13.
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
14.
Prism (geometry)
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In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism
15.
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
16.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
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Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
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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
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Pencil
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Pencils create marks by physical abrasion, leaving behind a trail of solid core material that adheres to a sheet of paper or other surface. They are distinct from pens, which instead disperse a trail of liquid or gel ink that stains the light colour of the paper, most pencil cores are made of graphite mixed with a clay binder which leaves grey or black marks that can be easily erased. Other types of core are less widely used, such as charcoal pencils. Grease pencils have a softer, crayon-like waxy core that can leave marks on smooth surfaces such as glass or porcelain, the most common type of pencil casing is of thin wood, usually hexagonal in section but sometimes cylindrical, permanently bonded to the core. Similar permanent casings may be constructed of materials such as plastic or paper. To use the pencil, the casing must be carved or peeled off to expose the working end of the core as a sharp point, mechanical pencils have more elaborate casings which support mobile pieces of pigment core that can be extended or retracted through the casing tip as needed. Pencil, from Old French pincel, from Latin penicillus a little tail originally referred to a fine brush of camel hair. Though the archetypal pencil was a brush, the stylus, a thin metal stick used for scratching in papyrus or wax tablets, was used extensively by the Romans. The meaning of writing implement apparently evolved late in the 16th century. Prior to 1565, a deposit of graphite was discovered on the approach to Grey Knotts from the hamlet of Seathwaite in Borrowdale parish, Cumbria. This particular deposit of graphite was pure and solid. This remains the only deposit of graphite ever found in this solid form. Chemistry was in its infancy and the substance was thought to be a form of lead. Many people have the misconception that the graphite in the pencil is lead, the words for pencil in German, Irish, Arabic, and other languages literally mean lead pen. The value of graphite was soon realised to be enormous, mainly because it could be used to line the moulds for cannonballs, when sufficient stores of graphite had been accumulated, the mines were flooded to prevent theft until more was required. Graphite had to be smuggled out for use in pencils, because graphite is soft, it requires some form of encasement. Graphite sticks were initially wrapped in string or sheepskin for stability, the news of the usefulness of these early pencils spread far and wide, attracting the attention of artists all over the known world. England continued to enjoy a monopoly on the production of pencils until a method of reconstituting the graphite powder was found, the distinctively square English pencils continued to be made with sticks cut from natural graphite into the 1860s
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Semiregular polyhedron
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The term semiregular polyhedron is used variously by different authors. In its original definition, it is a polyhedron with faces and a symmetry group which is transitive on its vertices. These polyhedra include, The thirteen Archimedean solids, an infinite series of convex prisms. An infinite series of convex antiprisms and these semiregular solids can be fully specified by a vertex configuration, a listing of the faces by number of sides in order as they occur around a vertex. For example,3.5.3.5, represents the icosidodecahedron which alternates two triangles and two pentagons around each vertex,3.3.3.5 in contrast is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive, since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte provided a definition which Coxeter found too artificial, Coxeter himself dubbed Gossets figures uniform, with only a quite restricted subset classified as semiregular. Yet others have taken the path, categorising more polyhedra as semiregular. These include, Three sets of polyhedra which meet Gossets definition. The duals of the above semiregular solids, arguing that since the polyhedra share the same symmetries as the originals. These duals include the Catalan solids, the convex dipyramids and antidipyramids or trapezohedra, a further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing. Gossets definition of semiregular includes figures of higher symmetry, the regular and quasiregular polyhedra and this naming system works well, and reconciles many of the confusions. Assuming that ones stated definition applies only to convex polyhedra is probably the most common failing, Coxeter, Cromwell and Cundy & Rollett are all guilty of such slips. In many works semiregular polyhedron is used as a synonym for Archimedean solid and we can distinguish between the facially-regular and vertex-transitive figures based on Gosset, and their vertically-regular and facially-transitive duals. Later, Coxeter would quote Gossets definition without comment, thus accepting it by implication, peter Cromwell writes in a footnote to Page 149 that, in current terminology, semiregular polyhedra refers to the Archimedean and Catalan solids. On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans, by implication this treats the Catalans as not semiregular, thus effectively contradicting the definition he provided in the earlier footnote. Semiregular polytope Regular polyhedron Weisstein, Eric W. Semiregular polyhedron
21.
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
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Hosohedron
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In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians, the restriction m ≥3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a tiling, this restriction may be relaxed, since digons can be represented as spherical lunes. Allowing m =2 admits a new class of regular polyhedra. On a spherical surface, the polyhedron is represented as n abutting lunes, all these lunes share two common vertices. The digonal faces of a 2n-hosohedron, represents the fundamental domains of symmetry in three dimensions, Cnv, order 2n. The reflection domains can be shown as alternately colored lunes as mirror images, bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n. The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the dual of the n-gonal hosohedron is the n-gonal dihedron. The polyhedron is self-dual, and is both a hosohedron and a dihedron, a hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism, in the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation, Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has two vertices, each with a vertex figure, the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M, Coxeter, and possibly derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”. Polyhedron Polytope McMullen, Peter, Schulte, Egon, Abstract Regular Polytopes, Cambridge University Press, ISBN 0-521-81496-0 Coxeter, H. S. M, ISBN 0-486-61480-8 Weisstein, Eric W. Hosohedron
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Cartesian product
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In Set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B, products can be specified using set-builder notation, e. g. A table can be created by taking the Cartesian product of a set of rows, If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More generally, a Cartesian product of n sets, also known as an n-fold Cartesian product, can be represented by an array of n dimensions, an ordered pair is a 2-tuple or couple. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, an illustrative example is the standard 52-card deck. The standard playing card ranks form a 13-element set, the card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form, both sets are distinct, even disjoint. The main historical example is the Cartesian plane in analytic geometry, usually, such a pairs first and second components are called its x and y coordinates, respectively, cf. picture. The set of all such pairs is thus assigned to the set of all points in the plane, a formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is =, note that, under this definition, X × Y ⊆ P, where P represents the power set. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, let A, B, C, and D be sets. × C ≠ A × If for example A =, then × A = ≠ = A ×, the Cartesian product behaves nicely with respect to intersections, cf. left picture. × = ∩ In most cases the above statement is not true if we replace intersection with union, cf. middle picture. Other properties related with subsets are, if A ⊆ B then A × C ⊆ B × C, the cardinality of a set is the number of elements of the set. For example, defining two sets, A = and B =, both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements, each element of A is paired with each element of B. Each pair makes up one element of the output set, the number of values in each element of the resulting set is equal to the number of sets whose cartesian product is being taken,2 in this case
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Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
25.
Hexagonal bipyramid
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A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces,8 vertices and 18 edges, the 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces and it is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is associated with the regular polyhedral form with pentagonal faces. The term dodecadeltahedron is sometimes used to distinguish the bipyramid from the Platonic solid, the hexagonal bipyramid has a plane of symmetry where the bases of the two pyramids are joined. This plane is a regular hexagon, there are also six planes of symmetry crossing through the two apices. These planes are rhombic and lie at 30° angles to each other, with an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. Hexagonal trapezohedron A similar 12-sided polyhedron with a twist and kite faces, snub disphenoid Another 12-sided polyhedron with 2-fold symmetry and only triangular faces. Archived from the original on 4 February 2007, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model hexagonal dipyramid Conway Notation for Polyhedra Try, dP6
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Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
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Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
28.
Uniform convex honeycomb
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In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. They can be considered the three-dimensional analogue to the uniform tilings of the plane, the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. 1905, Alfredo Andreini enumerated 25 of these tessellations,1991, Norman Johnsons manuscript Uniform Polytopes identified the complete list of 28. 1994, Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28 and he found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991, alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. Only 14 of the uniform polyhedra appear in these patterns. This set can be called the regular and semiregular honeycombs and it has been called the Archimedean honeycombs by analogy with the convex uniform polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations, the individual honeycombs are listed with names given to them by Norman Johnson. For cross-referencing, they are given with list indices from Andreini, Williams, Johnson, and Grünbaum. Coxeter uses δ4 for a honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb. The fundamental infinite Coxeter groups for 3-space are, The C ~3, cubic, The B ~3, alternated cubic, The A ~3 cyclic group, or, There is a correspondence between all three families. Removing one mirror from C ~3 produces B ~3 and this allows multiple constructions of the same honeycombs. If cells are colored based on positions within each Wythoff construction. In addition there are 5 special honeycombs which dont have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations, the total unique honeycombs above are 18. The total unique honeycombs above are 10. Combining these counts,18 and 10 gives us the total 28 uniform honeycombs, the regular cubic honeycomb, represented by Schläfli symbol, offers seven unique derived uniform honeycombs via truncation operations. The reflectional symmetry is the affine Coxeter group, There are four index 2 subgroups that generate alternations, and +, with the first two generated repeated forms, and the last two are nonuniform. The B ~4, group offers 11 derived forms via truncation operations, There are 3 index 2 subgroups that generate alternations, and +
29.
Hexagonal prismatic honeycomb
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The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms and it is constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms. It is constructed from a hexagonal tiling extruded into prisms and it is one of 28 convex uniform honeycombs. This honeycomb can be alternated into the gyrated tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps, the trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,2 and it is constructed from a trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated hexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1,2 and it is constructed from a truncated hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1,3,2 and it is constructed from a rhombitrihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the snub hexagonal prismatic honeycomb or simo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,8 and it is constructed from a snub hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated trihexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1,2,3 and it is constructed from a truncated trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1,2 and it is constructed from an elongated triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex and it can be seen as parallel planes of square tiling with alternating offsets caused by layers of paired triangular prisms
30.
Triangular-hexagonal prismatic honeycomb
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The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms and it is constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms. It is constructed from a hexagonal tiling extruded into prisms and it is one of 28 convex uniform honeycombs. This honeycomb can be alternated into the gyrated tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps, the trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,2 and it is constructed from a trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated hexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1,2 and it is constructed from a truncated hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1,3,2 and it is constructed from a rhombitrihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the snub hexagonal prismatic honeycomb or simo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,8 and it is constructed from a snub hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated trihexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1,2,3 and it is constructed from a truncated trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1,2 and it is constructed from an elongated triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex and it can be seen as parallel planes of square tiling with alternating offsets caused by layers of paired triangular prisms
31.
Snub triangular-hexagonal prismatic honeycomb
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The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms and it is constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms. It is constructed from a hexagonal tiling extruded into prisms and it is one of 28 convex uniform honeycombs. This honeycomb can be alternated into the gyrated tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps, the trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,2 and it is constructed from a trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated hexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1,2 and it is constructed from a truncated hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1,3,2 and it is constructed from a rhombitrihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the snub hexagonal prismatic honeycomb or simo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,8 and it is constructed from a snub hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated trihexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1,2,3 and it is constructed from a truncated trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1,2 and it is constructed from an elongated triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex and it can be seen as parallel planes of square tiling with alternating offsets caused by layers of paired triangular prisms
32.
Rhombitriangular-hexagonal prismatic honeycomb
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The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms and it is constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms. It is constructed from a hexagonal tiling extruded into prisms and it is one of 28 convex uniform honeycombs. This honeycomb can be alternated into the gyrated tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps, the trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,2 and it is constructed from a trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated hexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1,2 and it is constructed from a truncated hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1,3,2 and it is constructed from a rhombitrihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the snub hexagonal prismatic honeycomb or simo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,8 and it is constructed from a snub hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated trihexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1,2,3 and it is constructed from a truncated trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1,2 and it is constructed from an elongated triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex and it can be seen as parallel planes of square tiling with alternating offsets caused by layers of paired triangular prisms
33.
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
34.
Truncated tetrahedral prism
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In geometry, a truncated tetrahedral prism is a convex uniform polychoron. This polychoron has 10 polyhedral cells,2 truncated tetrahedra connected by 4 triangular prisms and 4 hexagonal prisms and it has 24 faces,8 triangular,18 square, and 8 hexagons. It has 48 edges and 24 vertices and it is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids. Net Truncated-tetrahedral dyadic prism Tuttip Truncated tetrahedral hyperprism 6, convex uniform prismatic polychora - Model 49, George Olshevsky. 4D uniform polytopes x x3x3o - tuttip
35.
Truncated octahedral prism
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In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells It has 64 faces, and 96 edges and 48 vertices and it has two symmetry constructions, one from the truncated octahedron, and one as an omnitruncation of the tetrahedron. It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids, the snub tetrahedral prism has symmetry, order 24, although as an icosahedral prism, its full symmetry is, order 240. It can also be seen as a truncated octahedral prism. It has 2 icosahedra connected by 6 tetrahedra and 8 octahedra, in total it has 40 cells,112 triangular faces,96 edges, and 24 vertices. It has symmetry, order 48, and also + symmetry, vertex figure for full snub tetrahedral antiprism Truncated 16-cell,6. Convex uniform prismatic polychora - Model 54, George Olshevsky, 4D uniform polytopes x x3x3x - tope
36.
Truncated cuboctahedral prism
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In geometry, a truncated cuboctahedral prism or great rhombicuboctahedral prism is a convex uniform polychoron. It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes. It has 76 cells,2 snub cubes connected by 12 tetrahedrons,6 square antiprisms, there are 48 vertices,192 edges, and 220 faces. It has + symmetry, order 48, vertex figure for full snub cuboctahedral antiprism 6. Convex uniform prismatic polychora - Model 55, George Olshevsky, 4D uniform polytopes x3x4x x - gircope
37.
Truncated icosahedral prism
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In geometry, a truncated icosahedral prism is a convex uniform polychoron. It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes, Truncated-icosahedral dyadic prism Tipe Truncated-icosahedral hyperprism Truncated 600-cell,6. Convex uniform prismatic polychora - Model 62, George Olshevsky, 4D uniform polytopes x x3o5x - tipe
38.
Truncated icosidodecahedral prism
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In geometry, a truncated icosidodecahedral prism or great rhombicosidodecahedral prism is a convex uniform 4-polytope. It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes. It has 184 cells,2 snub dodecahedrons connected by 30 tetrahedrons,12 pentagonal antiprisms and it has 120 vertices,220 edges, and 284 faces. It has + symmetry, order 120, vertex figure for full snub dodecahedral antiprism 6. Convex uniform prismatic polychora - Model 63, George Olshevsky, 4D uniform polytopes x x3o5x - griddip
39.
Runcitruncated 5-cell
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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
40.
Omnitruncated 5-cell
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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
41.
Runcitruncated 16-cell
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In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract. There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations, the runcinated tesseract or disprismatotesseractihexadecachoron has 16 tetrahedra,32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes,3 triangular prisms, the runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra, cubes, and triangular prisms. The same process applied to a 16-cell also yields the same figure, the other 24 cubical cells are connected to the former 8 cells via only two opposite square faces, the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces, the runcinated tesseract can be dissected into 2 cubic cupolae and a rhombicuboctahedral prism between them. This dissection can be analogous to the 3D rhombicuboctahedron being dissected into two square cupola and a central octagonal prism. The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a rhombicuboctahedral envelope, the images of its cells are laid out within this envelope as follows, The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope. Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron and these are the images of 12 of the cubical cells. The 18 square faces of the envelope are the images of the cubical cells. The 12 wedge-shaped volumes connecting the edges of the cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms. The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms, finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra. This layout of cells in projection is analogous to the layout of the faces of the rhombicuboctahedron under projection to 2 dimensions, the rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron, the runcitruncated tesseract, runcicantellated 16-cell, or prismatorhombated hexadecachoron is bounded by 80 cells,8 truncated cubes,16 cuboctahedra,24 octagonal prisms, and 32 triangular prisms. The runcitruncated tesseract may be constructed from the tesseract by expanding the truncated cube cells outward radially. In the process, the tetrahedra expand into cuboctahedra, and triangular prisms fill in the remaining gaps, two of the truncated cube cells project to a truncated cube in the center of the projection envelope. Six octagonal prisms connect this central truncated cube to the faces of the envelope. These are the images of 12 of the octahedral prism cells, the remaining 12 octahedral prisms are projected to the rectangular faces of the envelope. The 6 square faces of the envelope are the images of the remaining 6 truncated cube cells, twelve right-angle triangular prisms connect the inner octagonal prisms
42.
Omnitruncated tesseract
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In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract. There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations, the runcinated tesseract or disprismatotesseractihexadecachoron has 16 tetrahedra,32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes,3 triangular prisms, the runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra, cubes, and triangular prisms. The same process applied to a 16-cell also yields the same figure, the other 24 cubical cells are connected to the former 8 cells via only two opposite square faces, the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces, the runcinated tesseract can be dissected into 2 cubic cupolae and a rhombicuboctahedral prism between them. This dissection can be analogous to the 3D rhombicuboctahedron being dissected into two square cupola and a central octagonal prism. The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a rhombicuboctahedral envelope, the images of its cells are laid out within this envelope as follows, The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope. Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron and these are the images of 12 of the cubical cells. The 18 square faces of the envelope are the images of the cubical cells. The 12 wedge-shaped volumes connecting the edges of the cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms. The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms, finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra. This layout of cells in projection is analogous to the layout of the faces of the rhombicuboctahedron under projection to 2 dimensions, the rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron, the runcitruncated tesseract, runcicantellated 16-cell, or prismatorhombated hexadecachoron is bounded by 80 cells,8 truncated cubes,16 cuboctahedra,24 octagonal prisms, and 32 triangular prisms. The runcitruncated tesseract may be constructed from the tesseract by expanding the truncated cube cells outward radially. In the process, the tetrahedra expand into cuboctahedra, and triangular prisms fill in the remaining gaps, two of the truncated cube cells project to a truncated cube in the center of the projection envelope. Six octagonal prisms connect this central truncated cube to the faces of the envelope. These are the images of 12 of the octahedral prism cells, the remaining 12 octahedral prisms are projected to the rectangular faces of the envelope. The 6 square faces of the envelope are the images of the remaining 6 truncated cube cells, twelve right-angle triangular prisms connect the inner octagonal prisms
43.
Runcitruncated 24-cell
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In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell. There are 3 unique degrees of runcinations of the 24-cell including with permutations truncations and cantellations, in geometry, the runcinated 24-cell or small prismatotetracontoctachoron is a uniform 4-polytope bounded by 48 octahedra and 192 triangular prisms. The octahedral cells correspond with the cells of a 24-cell and its dual, E. L. Elte identified it in 1912 as a semiregular polytope. The regular skew polyhedron, exists in 4-space with 8 square around each vertex and these square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. The 384 triangular faces of the runcinated 24-cell can be seen as removed, the dual regular skew polyhedron, is similarly related to the octagonal faces of the bitruncated 24-cell. The runcitruncated 24-cell or prismatorhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell, like the snub 24-cell, it has symmetry, order 576. The runcitruncated 24-cell has 192 identical hexagonal faces, while the runcicantic snub 24-cell has 2 constructive sets of 96 hexagons, the difference can be seen in the vertex figures, A related 4-polytope is the runcic snub 24-cell or prismatorhombisnub icositetrachoron, s3. It is not uniform, but it is vertex-transitive and has all regular polygon faces. It is constructed with 24 icosahedra,24 truncated tetrahedra,96 triangular prisms, and 96 triangular cupolae in the gaps, for a total of 240 cells,960 faces,1008 edges, like the snub 24-cell, it has symmetry, order 576. The vertex figure contains one icosahedron, two triangular prisms, one truncated tetrahedron, and 3 triangular cupolae, the omnitruncated 24-cell or great prismatotetracontoctachoron is a uniform 4-polytope derived from the 24-cell. It is composed of 1152 vertices,2304 edges, and 1392 faces and it has 240 cells,48 truncated cuboctahedra,192 hexagonal prisms. Each vertex contains four cells in a tetrahedral vertex figure. The 48 great rhombicuboctahedral cells are joined to each other via their octagonal faces and they can be grouped into two groups of 24 each, corresponding with the cells of a 24-cell and its dual. Its vertex figure contains 4 tetrahedra,2 octahedra, and 2 snub cubes and it has 816 cells,2832 faces,2592 edges, and 576 vertices. 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 Four-dimensional Archimedean Polytopes, Marco Möller,2004 PhD dissertation m58 m59 m533
44.
Omnitruncated 24-cell
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In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell. There are 3 unique degrees of runcinations of the 24-cell including with permutations truncations and cantellations, in geometry, the runcinated 24-cell or small prismatotetracontoctachoron is a uniform 4-polytope bounded by 48 octahedra and 192 triangular prisms. The octahedral cells correspond with the cells of a 24-cell and its dual, E. L. Elte identified it in 1912 as a semiregular polytope. The regular skew polyhedron, exists in 4-space with 8 square around each vertex and these square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. The 384 triangular faces of the runcinated 24-cell can be seen as removed, the dual regular skew polyhedron, is similarly related to the octagonal faces of the bitruncated 24-cell. The runcitruncated 24-cell or prismatorhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell, like the snub 24-cell, it has symmetry, order 576. The runcitruncated 24-cell has 192 identical hexagonal faces, while the runcicantic snub 24-cell has 2 constructive sets of 96 hexagons, the difference can be seen in the vertex figures, A related 4-polytope is the runcic snub 24-cell or prismatorhombisnub icositetrachoron, s3. It is not uniform, but it is vertex-transitive and has all regular polygon faces. It is constructed with 24 icosahedra,24 truncated tetrahedra,96 triangular prisms, and 96 triangular cupolae in the gaps, for a total of 240 cells,960 faces,1008 edges, like the snub 24-cell, it has symmetry, order 576. The vertex figure contains one icosahedron, two triangular prisms, one truncated tetrahedron, and 3 triangular cupolae, the omnitruncated 24-cell or great prismatotetracontoctachoron is a uniform 4-polytope derived from the 24-cell. It is composed of 1152 vertices,2304 edges, and 1392 faces and it has 240 cells,48 truncated cuboctahedra,192 hexagonal prisms. Each vertex contains four cells in a tetrahedral vertex figure. The 48 great rhombicuboctahedral cells are joined to each other via their octagonal faces and they can be grouped into two groups of 24 each, corresponding with the cells of a 24-cell and its dual. Its vertex figure contains 4 tetrahedra,2 octahedra, and 2 snub cubes and it has 816 cells,2832 faces,2592 edges, and 576 vertices. 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 Four-dimensional Archimedean Polytopes, Marco Möller,2004 PhD dissertation m58 m59 m533
45.
Runcitruncated 600-cell
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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
46.
Omnitruncated 120-cell
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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
47.
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
48.
Dihedron
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A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons, a regular dihedron is the dihedron formed by two regular polygons, which may be described by the Schläfli symbol. As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere, the dual of a n-gonal dihedron is the n-gonal hosohedron, where n digon faces share two vertices. A dihedron can be considered a degenerate prism consisting of two n-sided polygons connected back-to-back, so that the object has no depth. The polygons must be congruent, but glued in such a way one is the mirror image of the other. This characterization holds also for the distances on the surface of a dihedron, as a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. The regular polyhedron is self-dual, and is both a hosohedron and a dihedron, in the limit the dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation, A regular ditope is an n-dimensional analogue of a dihedron, with Schläfli symbol. It has two facets, which share all ridges, in common, polyhedron Polytope Weisstein, Eric W. Dihedron
49.
Hexagonal antiprism
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In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to other. In the case of a regular 6-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, as faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles. If faces are all regular, it is a semiregular polyhedron, the hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles. Hexagonal Antiprism, Interactive Polyhedron model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A6