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.
Octagonal dipyramid
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The octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles and it can be drawn as a tiling on a sphere which also represents the fundamental domains of, *422 symmetry, Weisstein, Eric W. Dipyramid. Archived from the original on 4 February 2007, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <8> Conway Notation for Polyhedra Try, dP8

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.
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

16.
Optics
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Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light, because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for using the classical description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice, practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines, physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that waves were in fact electromagnetic radiation. Some phenomena depend on the fact that light has both wave-like and particle-like properties, explanation of these effects requires quantum mechanics. When considering lights particle-like properties, the light is modelled as a collection of particles called photons, quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy, various engineering fields, photography, practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, lasers, and fibre optics. Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, the earliest known lenses, made from polished crystal, often quartz, date from as early as 700 BC for Assyrian lenses such as the Layard/Nimrud lens. The ancient Romans and Greeks filled glass spheres with water to make lenses, the word optics comes from the ancient Greek word ὀπτική, meaning appearance, look. Greek philosophy on optics broke down into two opposing theories on how vision worked, the theory and the emission theory. The intro-mission approach saw vision as coming from objects casting off copies of themselves that were captured by the eye, plato first articulated the emission theory, the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus, some hundred years later, Euclid wrote a treatise entitled Optics where he linked vision to geometry, creating geometrical optics. Ptolemy, in his treatise Optics, held a theory of vision, the rays from the eye formed a cone, the vertex being within the eye. The rays were sensitive, and conveyed back to the observer’s intellect about the distance. He summarised much of Euclid and went on to describe a way to measure the angle of refraction, during the Middle Ages, Greek ideas about optics were resurrected and extended by writers in the Muslim world

17.
Movie projector
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A movie projector is an opto-mechanical device for displaying motion picture film by projecting it onto a screen. Most of the optical and mechanical elements, except for the illumination, the first movie projector was the Zoopraxiscope, invented by British photographer Eadweard Muybridge in 1879. The zoopraxiscope projected images from rotating glass disks in rapid succession to give the impression of motion, the stop-motion images were initially painted onto the glass, as silhouettes. A second series of discs, made in 1892–94, used outline drawings printed onto the discs photographically, a more sophisticated movie projector was invented by Frenchman Louis Le Prince while working in Leeds. In 1888 Le Prince took out a patent for a 16-lens device that combined a motion picture camera with a projector, in 1888, he used an updated version of his camera to film the first ever motion picture, the Roundhay Garden Scene. The pictures were exhibited in Hunslet. The Lumière brothers invented the first successful movie projector and they made their first film, Sortie de lusine Lumière de Lyon, in 1894, which was publicly screened at LEden, La Ciotat a year later. The first commercial, public screening of cinematographic films happened in Paris on 28 December 1895, the cinematograph was also exhibited at the Paris Exhibition of 1900. At the Exhibition, films made by the Lumière Brothers were projected onto a screen measuring 16 by 21 meters. In 1999, digital projectors were being tried out in some movie theatres. These early projectors played the movie stored on a server and played back through the projector, due to their relatively low resolution, the images at the time showed pixelization blocks in some scenes, much like images on early widescreen televisions. By 2006, the advent of much higher 4K resolution digital projection had removed any traces of pixelization, the systems became more compact than the larger machines of four years earlier. By 2009, movie theatres started replacing the film projectors with digital projectors, in 2013, it was estimated that 92% of movie theatres in the United States had converted to digital, with 8% still playing film. In 2015, numerous popular filmmakers—including Quentin Tarantino and Christopher Nolan—lobbied large studios to commit to purchase an amount of 35 mm film from Kodak. The decision ensured that Kodaks 35mm film production would continue for several years, nowadays film projectors are considered obsolete as high-resolution digital projectors offer many advantages over traditional film units. For example, digital projectors contain no moving parts except fans, can be operated remotely and they also allow for much easier, less expensive, and more reliable storage and distribution of content, including the ability to display live broadcasts. According to the theory of the phi phenomenon, the brain constitutes an experience of apparent movement when presented with a sequence of near-identical still images. Persistence of vision should be compared with the phenomena of beta movement

18.
Uniform honeycomb
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In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and its dimension can be clarified as n-honeycomb for an n-dimensional honeycomb. An n-dimensional uniform honeycomb can be constructed on the surface of n-spheres, in n-dimensional Euclidean space, a 2-dimensional uniform honeycomb is more often called a uniform tiling or uniform tessellation. Nearly all uniform tessellations can be generated by a Wythoff construction, wythoffian tessellations can be defined by a vertex figure. For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex, for example 4.4.4.4 represents a regular tessellation, a square tiling, with 4 squares around each vertex. Norman Johnson Uniform Polytopes, Manuscript Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. CS1 maint, Multiple names, authors list H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Critchlow, order in Space, A design source book. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative, Mem. Società Italiana della Scienze, Ser.3,14 75–129, tessellations of the Plane Klitzing, Richard

19.
Truncated square prismatic honeycomb
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The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex and its vertex figure is a regular octahedron. It is a tessellation with Schläfli symbol. John Horton Conway calls this honeycomb a cubille, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. The Cartesian coordinates of the vertices are, for all values, i, j, k, with edges parallel to the axes. It is part of a family of hypercube honeycombs, with Schläfli symbols of the form, starting with the square tiling. It is one of 28 uniform honeycombs using convex uniform polyhedral cells, simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems, There is a large number of uniform colorings, derived from different symmetries. These include, It is related to the regular 4-polytope tesseract, Schläfli symbol, which exists in 4-space and its also related to the order-5 cubic honeycomb, Schläfli symbol, of hyperbolic space with 5 cubes around each edge. It is in a sequence of polychora and honeycomb with octahedral vertex figures and it in a sequence of regular polytopes and honeycombs with cubic cells. The, Coxeter group generates 15 permutations of uniform tessellations,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform tessellations,4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the A ~3 Coxeter group and it is composed of octahedra and cuboctahedra in a ratio of 1,1. John Horton Conway calls this honeycomb a cuboctahedrille, and its dual oblate octahedrille, There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below. This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra and this scaliform honeycomb is represented by Coxeter diagram, and symbol s3, with coxeter notation symmetry. The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space and it is composed of truncated cubes and octahedra in a ratio of 1,1

20.
Omnitruncated cubic honeycomb
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The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex and its vertex figure is a regular octahedron. It is a tessellation with Schläfli symbol. John Horton Conway calls this honeycomb a cubille, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. The Cartesian coordinates of the vertices are, for all values, i, j, k, with edges parallel to the axes. It is part of a family of hypercube honeycombs, with Schläfli symbols of the form, starting with the square tiling. It is one of 28 uniform honeycombs using convex uniform polyhedral cells, simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems, There is a large number of uniform colorings, derived from different symmetries. These include, It is related to the regular 4-polytope tesseract, Schläfli symbol, which exists in 4-space and its also related to the order-5 cubic honeycomb, Schläfli symbol, of hyperbolic space with 5 cubes around each edge. It is in a sequence of polychora and honeycomb with octahedral vertex figures and it in a sequence of regular polytopes and honeycombs with cubic cells. The, Coxeter group generates 15 permutations of uniform tessellations,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform tessellations,4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the A ~3 Coxeter group and it is composed of octahedra and cuboctahedra in a ratio of 1,1. John Horton Conway calls this honeycomb a cuboctahedrille, and its dual oblate octahedrille, There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below. This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra and this scaliform honeycomb is represented by Coxeter diagram, and symbol s3, with coxeter notation symmetry. The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space and it is composed of truncated cubes and octahedra in a ratio of 1,1

21.
Runcitruncated cubic honeycomb
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The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex and its vertex figure is a regular octahedron. It is a tessellation with Schläfli symbol. John Horton Conway calls this honeycomb a cubille, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. The Cartesian coordinates of the vertices are, for all values, i, j, k, with edges parallel to the axes. It is part of a family of hypercube honeycombs, with Schläfli symbols of the form, starting with the square tiling. It is one of 28 uniform honeycombs using convex uniform polyhedral cells, simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems, There is a large number of uniform colorings, derived from different symmetries. These include, It is related to the regular 4-polytope tesseract, Schläfli symbol, which exists in 4-space and its also related to the order-5 cubic honeycomb, Schläfli symbol, of hyperbolic space with 5 cubes around each edge. It is in a sequence of polychora and honeycomb with octahedral vertex figures and it in a sequence of regular polytopes and honeycombs with cubic cells. The, Coxeter group generates 15 permutations of uniform tessellations,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform tessellations,4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the A ~3 Coxeter group and it is composed of octahedra and cuboctahedra in a ratio of 1,1. John Horton Conway calls this honeycomb a cuboctahedrille, and its dual oblate octahedrille, There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below. This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra and this scaliform honeycomb is represented by Coxeter diagram, and symbol s3, with coxeter notation symmetry. The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space and it is composed of truncated cubes and octahedra in a ratio of 1,1

22.
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

23.
Runcitruncated 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

24.
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

25.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950

26.
Triangular prism
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In geometry, a triangular prism is a three-sided prism, it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique, a uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the normals of the other three are in the same plane. All cross-sections parallel to the faces are the same triangle. A right triangular prism is semiregular or, more generally, a uniform if the base faces are equilateral triangles. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t, alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product x. The dual of a prism is a triangular bipyramid. The symmetry group of a right 3-sided prism with triangular base is D3h of order 12, the rotation group is D3 of order 6. The symmetry group does not contain inversion, the volume of any prism is the product of the area of the base and the distance between the two bases. A truncated right triangular prism has one triangular face truncated at an oblique angle, there are two full D2h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C3v symmetry faceting have one triangle,3 lateral crossed square faces. This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations and this polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure, and continues as tilings of the hyperbolic plane. These vertex-transitive figures have reflectional symmetry and this polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure, and continues as tilings of the hyperbolic plane. These vertex-transitive figures have reflectional symmetry, there are 4 uniform compounds of triangular prisms, Compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms. Each progressive uniform polytope is constructed vertex figure of the previous polytope, thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. In Coxeters notation the triangular prism is given the symbol −121, the triangular prism exists as cells of a number of four-dimensional uniform 4-polytopes, including, Wedge Weisstein, Eric W. Triangular prism. Interactive Polyhedron, Triangular Prism surface area and volume of a triangular prism

27.
Cube
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Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0. 5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the lungs and the bloodstream, the International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen, beryllium copper is a non-ferrous alloy used in springs, spring wire, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves

28.
Pentagonal prism
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In geometry, the pentagonal prism a prism with a pentagonal base. It is a type of heptahedron with 7 faces,15 edges and it can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a pentagon and a line segment. The dual of a prism is a pentagonal bipyramid. The symmetry group of a pentagonal prism is D5h of order 20. The rotation group is D5 of order 10, the volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. It exists as cells of four nonprismatic uniform 4-polytopes in 4 dimensions, Weisstein, Pentagonal Prism Polyhedron Model -- works in your web browser

29.
Hexagonal prism
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In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces,18 edges, and 12 vertices, since it has eight faces, it is an octahedron. However, the octahedron is primarily used to refer to the regular octahedron. Because of the ambiguity of the octahedron and the dissimilarity of the various eight-sided figures. Before sharpening, many take the shape of a long hexagonal prism. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t, alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product ×. The dual of a prism is a hexagonal bipyramid. The symmetry group of a hexagonal prism is D6h of order 24. The rotation group is D6 of order 12, for p <6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p >6, they are tilings of the hyperbolic plane, Uniform Honeycombs in 3-Space VRML models The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms Weisstein, Eric W. Hexagonal prism. Hexagonal Prism Interactive Model -- works in your web browser

30.
Decagonal prism
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In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra, the decagonal prism has 12 faces,30 edges, and 20 vertices. If faces are all regular, it is a semiregular or prismatic uniform polyhedron, the decagonal prism exists as cells in two four-dimensional uniform 4-polytopes, Weisstein, Eric W. Prism. 3-d model of a Decagonal Prism

31.
List of planar symmetry groups
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This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes, International notation, orbifold notation, and Coxeter notation. There are three kinds of groups of the plane,2 rosette groups – 2D point groups 7 frieze groups – 2D line groups 17 wallpaper groups – 2D space groups. There are two families of discrete point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. The 7 frieze groups, the line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups, the yellow regions represent the infinite fundamental domain in each. The p1 and p2 groups, with no symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Coxeter, H. S. M. & Moser, W. O. J. Generators and Relations for Discrete Groups. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Conways manuscript on Orbifold notation The 17 Wallpaper Groups

32.
Truncated cuboctahedron
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In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces,8 regular hexagonal faces,6 regular octagonal faces,48 vertices and 72 edges, since each of its faces has point symmetry, the truncated cuboctahedron is a zonohedron. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure, however, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular. The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. One unfortunate point of confusion, There is a uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron.7551724 a 2 V = a 3 ≈41.7989899 a 3, many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupolas creates a genus 3 torus, There is only one uniform coloring of the faces of this polyhedron, one color for each face type. A 2-uniform coloring, with symmetry, exists with alternately colored hexagons. The truncated cuboctahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, the truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. This polyhedron can be considered a member of a sequence of patterns with vertex configuration. For p <6, the members of the sequence are omnitruncated polyhedra, for p <6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. In the mathematical field of theory, a truncated cuboctahedral graph is the graph of vertices and edges of the truncated cuboctahedron. It has 48 vertices and 72 edges, and is a zero-symmetric and cubic Archimedean graph, cube Cuboctahedron Octahedron Truncated icosidodecahedron Truncated octahedron – truncated tetratetrahedron Cromwell, P. Polyhedra. Eric W. Weisstein, Great rhombicuboctahedron at MathWorld, 3D convex uniform polyhedra x3x4x - girco

33.
Truncated square tiling
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In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular polygons which contains an octagon. It has Schläfli symbol of t. Conway calls it a truncated quadrille, other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges. There are 3 regular and 8 semiregular tilings in the plane, there are two distinct uniform colorings of a truncated square tiling. The truncated square tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing, since there is an even number of sides of all the polygons, the circles can be alternately colored as shown below. One variations on this pattern, often called a Mediterranean pattern, is shown in stone tiles with smaller squares, other variations stretch the squares or octagons. The Pythagorean tiling alternates large and small squares, and may be seen as identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices, in the plane it can be represented by a compound tiling, or combined can be seen as a chamfered square tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. However treating faces identically, there are three unique topologically forms, square tiling, truncated square tiling, snub square tiling. The tetrakis square tiling is the tiling of the Euclidean plane dual to the square tiling. It can be constructed square tiling with each divided into four isosceles right triangles from the center point. Conway calls it a kisquadrille, represented by a kis operation that adds a center point and it is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices. CS1 maint, Multiple names, authors list Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. Dale Seymour and Jill Britton, Introduction to Tessellations,1989, ISBN 978-0866514613, pp. 50–56 http, //www. decrete. com/stencils/octagontile Weisstein, 2D Euclidean tilings o4x4x - tosquat - O6

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Truncated tetrapentagonal tiling
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In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2 or tr, there are four small index subgroup constructed from by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, a radical subgroup is constructed, index 10, as, with gyration points removed, becoming orbifold, and its direct subgroup +, index 20, becomes orbifold. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch

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Truncated tetrahexagonal tiling
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In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex and it has Schläfli symbol of tr. From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full symmetry. The dual of the tiling represents the fundamental domains of orbifold symmetry, from symmetry, there are 15 small index subgroup by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images unique mirrors are colored red, green, and blue, the, subgroup has narrow lines representing glide reflections. The subgroup index-8 group, is the subgroup of. Larger subgroup constructed as, removing the gyration points of, index 6 becomes, finally their direct subgroups +, +, subgroup indices 12 and 24 respectively, can be given in orbifold notation as and. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch

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Truncated tetraheptagonal tiling
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In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr, Poincaré disk projection, centered on 14-gon, The dual to this tiling represents the fundamental domains of symmetry. There are 3 small index subgroups constructed from by mirror removal, in these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Chapter 10, the Beauty of Geometry, Twelve Essays. Uniform tilings in hyperbolic plane List of regular polytopes Weisstein, Eric W. Hyperbolic tiling, Weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch

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Truncated tetraoctagonal tiling
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In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex and it has Schläfli symbol of tr. There are 15 subgroups constructed from by mirror removal and alternation, mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met, in these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, is the subgroup of. A larger subgroup is constructed as, index 8, as, with gyration points removed, becomes or, and their direct subgroups +, +, subgroup indices 16 and 32 respectively, can be given in orbifold notation as and. From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full symmetry. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch

38.
Truncated tetraapeirogonal tiling
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In geometry, the truncated tetrapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex and it has Schläfli symbol of tr. The dual of this represents the fundamental domains of, symmetry. There are 15 small index subgroups constructed from by mirror removal, mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met, in these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, is the subgroup of. A larger subgroup is constructed as, index 8, as, with gyration points removed, becomes or, and their direct subgroups +, +, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as and. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk

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Octagonal bipyramid
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The octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles and it can be drawn as a tiling on a sphere which also represents the fundamental domains of, *422 symmetry, Weisstein, Eric W. Dipyramid. Archived from the original on 4 February 2007, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <8> Conway Notation for Polyhedra Try, dP8

40.
Disdyakis dodecahedron
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In geometry, a disdyakis dodecahedron, or hexakis octahedron or kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons, more formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. Its collective edges represent the reflection planes of the symmetry and it can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron. Seen in stereographic projection the edges of the dodecahedron form 9 circles in the plane. Between a polyhedron and its dual, vertices and faces are swapped in positions, the disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. It is a polyhedra in a sequence defined by the face configuration V4.6. 2n, 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. First stellation of rhombic dodecahedron Disdyakis triacontahedron Kisrhombille tiling Great rhombihexacron—A uniform dual polyhedron with the surface topology Williams. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Disdyakis dodecahedron at MathWorld

41.
Tetrakis square tiling
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In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each divided into four isosceles right triangles from the center point. Conway calls it a kisquadrille, represented by a kis operation that adds a center point and it is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices. It is labeled V4.8.8 because each isosceles triangle face has two types of vertices, one with 4 triangles, and two with 8 triangles and it is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex. A5 ×9 portion of the square tiling is used to form the board for the Malagasy board game Fanorona. In this game, pieces are placed on the vertices of the tiling, a similar board is also used for the Brazilian game Adugo, and for the game of Hare and Hounds. The tetrakis square tiling was used for a set of postage stamps issued by the United States Postal Service in 1997. This tiling also forms the basis for a commonly used pinwheel, windmill and these lines form the axes of symmetry of a reflection group, which has the triangles of the tiling as its fundamental domains. Rotational symmetry is shown by white and blue colored areas with a single fundamental domain for each subgroup is filled in yellow. Glide reflections are given with dashed lines, subgroups can be expressed as Coxeter diagrams, along with fundamental domain diagrams. It is topologically related to a series of polyhedra and tilings with face configuration Vn.6.6, tilings of regular polygons List of uniform tilings Percolation threshold Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. Keith Critchlow, Order in Space, A design source book,1970, p. 77-76, pattern 8

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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