1.
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

2.
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

3.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces

4.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski

5.
Octagonal prism
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In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron, the octagonal prism can also be seen as a tiling on a sphere, In optics, octagonal prisms are used to generate flicker-free images in movie projectors. It is an element of three uniform honeycombs, It is also an element of two four-dimensional uniform 4-polytopes, Weisstein, Eric W. Octagonal prism, interactive model of an Octagonal Prism

6.
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

7.
Square antiprism
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In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube, if all its faces are regular, it is a semiregular polyhedron or uniform polyhedron. One molecule with this geometry is the ion in the salt nitrosonium octafluoroxenate, however. Very few ions are cubical because such a shape would cause large repulsion between ligands, PaF3−8 is one of the few examples, in addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope. The main building block of the One World Trade Center has the shape of a tall tapering square antiprism. It is not a true antiprism because of its taper, the top square has half the area of the bottom one, a twisted prism can be made with the same vertex arrangement. It can be seen as the form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices and it has half of the symmetry of the uniform solution, Dn, +, order 8. The gyroelongated square pyramid is a Johnson solid constructed by augmenting one a square pyramid, similarly, the gyroelongated square bipyramid is a deltahedron constructed by replacing both squares of a square antiprism with a square pyramid. The snub disphenoid is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles, the snub square antiprism can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona and the sphenomegacorona are other Johnson solids that, like the square antiprism, the square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3. n. Compound of three square antiprisms Weisstein, Eric W. Antiprism, square Antiprism interactive model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A4

8.
Antiprism
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In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids and are a type of snub polyhedra, Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular n-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained when the line connecting the centers is perpendicular to the base planes. As faces, it has the two bases and, connecting those bases, 2n isosceles triangles. A uniform antiprism has, apart from the faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms form a series of vertex-uniform polyhedra. For n =2 we have as degenerate case the regular tetrahedron as a digonal antiprism, the dual polyhedra of the antiprisms are the trapezohedra. Let a be the edge-length of a uniform antiprism, then the volume is V = n 4 cos 2 π2 n −1 sin 3 π2 n 12 sin 2 π n a 3 and the surface area is A = n 2 a 2. There are a set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron. These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower form of the icosahedron. The symmetry group contains inversion if and only if n is odd, uniform star antiprisms are named by their star polygon bases, and exist in prograde and retrograde solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/ instead of p/q, in the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry. Some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, star antiprism compounds also can be constructed where p and q have common factors, thus a 10/4 antiprism is the compound of two 5/2 star antiprisms. Prism Apeirogonal antiprism Grand antiprism – a four-dimensional polytope One World Trade Center, California, University of California Press Berkeley. Chapter 2, Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Antiprism, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, nonconvex Prisms and Antiprisms Paper models of prisms and antiprisms

9.
Square cupola
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In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids. It can be obtained as a slice of the rhombicuboctahedron, as in all cupolae, the base polygon has twice as many edges and vertices as the top, in this case the base polygon is an octagon. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform and they were named by Norman Johnson, who first listed these polyhedra in 1966. The following formulae for volume, surface area, and circumradius can be used if all faces are regular and it can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, analogously to how the square cupola may be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the polygon has twice as many edges and vertices as the top. It may be seen as a cupola with a square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola. Eric W. Weisstein, Square cupola at MathWorld

10.
Johnson solid
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In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform, and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, an example of a Johnson solid is the square-based pyramid with equilateral sides, it has 1 square face and 4 triangular faces. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees, since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid is an example that actually has a degree-5 vertex. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3,4,5,6,8. In 1966, Norman Johnson published a list which included all 92 solids and he did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnsons list was complete, however, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid. The naming of Johnson Solids follows a flexible & precise descriptive formula, from there, a series of prefixes are attached to the word to indicate additions, rotations and transformations, Bi- indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, the solids can be joined so that like either faces or unlike faces meet, using this nomenclature, an octahedron can be described as a square bipyramid, a cuboctahedron as a triangular gyrobicupola, and an icosidodecahedron as a pentagonal gyrobirotunda. Elongated indicates a prism is joined to the base of the solid in question, a rhombicuboctahedron can thus be described as an elongated square orthobicupola. Gyroelongated indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids, an icosahedron can thus be described as a gyroelongated pentagonal bipyramid. Augmented indicates a pyramid or cupola is joined to one or more faces of the solid in question, diminished indicates a pyramid or cupola is removed from one or more faces of the solid in question. Gyrate indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, the last three operations — augmentation, diminution, and gyration — can be performed multiple times certain large solids. Bi- & Tri- indicate a double and treble operation respectively, for example, a bigyrate solid has two rotated cupolae, and a tridiminished solid has three removed pyramids or cupolae. In in certain solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. Para- indicates the former, that the solid in question has altered parallel faces, for example, a parabiaugmented solid has had two parallel faces augmented, and a metabigyrate solid has had 2 oblique faces gyrated. The last few Johnson solids have names based on certain polygon complexes from which they are assembled and these names are defined by Johnson with the following nomenclature, A lune is a complex of two triangles attached to opposite sides of a square. Spheno- indicates a complex formed by two adjacent lunes

11.
Pentagonal bipyramid
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In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids. Each bipyramid is the dual of a uniform prism, although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have five faces. If the faces are triangles, it is a deltahedron. It can be seen as two pentagonal pyramids connected by their bases, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, the pentagonal dipyramid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces, Pentagonal bipyramidal molecular geometry Eric W. Weisstein, Pentagonal dipyramid at MathWorld. Conway Notation for Polyhedra Try, dP5

12.
Bipyramid
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An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices, the referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. A right bipyramid has two points above and below the centroid of its base, nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is implied to be a right bipyramid. A right bipyramid can be represented as + P for internal polygon P, a concave bipyramid has a concave interior polygon. The face-transitive regular bipyramids are the dual polyhedra of the uniform prisms, a bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator. Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh, the volume of a bipyramid is V =2/3Bh where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the distance from the plane which contains the base. The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore, only three kinds of bipyramids can have all edges of the same length, the triangular, tetragonal, and pentagonal bipyramids. The rotation group is Dn of order 2n, except in the case of an octahedron, which has the larger symmetry group O of order 24. The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of symmetry in three dimensions, Dnh, order 4n. The reflection domains can be shown as alternately colored triangles as mirror images, a scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles. In one type the 2n vertices around the center alternate in rings above, in the other type, the 2n vertices are on the same plane, but alternate in two radii. The first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, in crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedra, the second has symmetry Dn, order 2n. The smallest scalenohedron has 8 faces and is identical to the regular octahedron. The second type is a rhombic bipyramid, the first type has 6 vertices can be represented as, where z is a parameter between 0 and 1, creating a regular octahedron at z =0, and becoming a disphenoid with merged coplanar faces at z =1. For z >1, it becomes concave, self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points

13.
Augmented pentagonal prism
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In geometry, the augmented pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid to one of its equatorial faces, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, weisstein, Eric W. Augmented pentagonal prism

14.
Pentagonal trapezohedron
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The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces which are congruent kites and it can be decomposed into two pentagonal pyramids and a pentagonal antiprism in the middle. It can also be decomposed into two pentagonal pyramids and a dodecahedron in the middle, the pentagonal trapezohedron was patented for use as a gaming die in 1906. Subsequent patents on ten-sided dice have made minor refinements to the design by rounding or truncating the edges. This enables the die to tumble so that the outcome is less predictable, one such refinement became notorious at the 1980 Gen Con when the patent was incorrectly thought to cover ten-sided dice in general. Ten-sided dice are commonly numbered from 0 to 9, as this allows two to be rolled in order to obtain a percentile result. Where one die represents the tens, the other represents units therefore a result of 7 on the former and 0 on the latter would be combined to produce 70, a result of double-zero is commonly interpreted as 100. Ten-sided dice may also be numbered 1 to 10 for use in games where a number in this range is desirable. A fairly consistent arrangement of the faces on ten-digit dice has been observed, the even and odd digits are divided among the two opposing caps of the die, and each pair of opposite faces adds to nine. When casting a 10-sided die, if numbered from 0-9, two are used to obtain a percentage roll. Rolling 2 of these are attributed in the results 00-99, where 00 can be viewed as a 100 as the result in some games. Alone casting a 0-9 ten sided dice, the 0 face is valued at 10, cundy H. M and Rollett, A. P. Mathematical models, 2nd Edn. Oxford University Press, p.117 Generalized formula of uniform polyhedron having 2n congruent right kite faces from Academia. edu Weisstein, virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, dA5

15.
Trapezohedron
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The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites, the n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces, an n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism. These figures, sometimes called deltohedra, must not be confused with deltahedra, in texts describing the crystal habits of minerals, the word trapezohedron is often used for the polyhedron properly known as a deltoidal icositetrahedron. In the case of the dual of a triangular antiprism the kites are rhombi and they are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces, a special case of a rhombohedron is one in the which the rhombi which form the faces have angles of 60° and 120°. It can be decomposed into two regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra, a degenerate form, n =2, form a geometric tetrahedron with 6 vertices,8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a form of antiprism, also a tetrahedron. The symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the symmetry group Od of order 48. The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, if the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n. Crystal arrangements of atoms can repeat in space with trapezohedral cells, the pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired, two dice of different colors are typically used for the two digits to represent numbers from 00 to 99. Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to two points. Diminished trapezohedron Rhombic dodecahedron Rhombic triacontahedron Bipyramid Conway polyhedron notation Anthony Pugh, California, University of California Press Berkeley. Chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <3> <4> <5> <6> <7> <8> <9> <10> Conway Notation for Polyhedra Try, dAn, where n=3,4,5. Example dA5 is a pentagonal trapezohedron

16.
Dice
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Dice are small throwable objects with multiple resting positions, used for generating random numbers. Dice are suitable as gambling devices for games like craps and are used in non-gambling tabletop games. A traditional die is a cube, with each of its six faces showing a different number of dots from 1 to 6. When thrown or rolled, the die comes to rest showing on its surface a random integer from one to six. A variety of devices are also described as dice, such specialized dice may have polyhedral or irregular shapes. They may be used to produce other than one through six. Loaded and crooked dice are designed to favor some results over others for purposes of cheating or amusement. A dice tray, a used to contain thrown dice, is sometimes used for gambling or board games. Dice have been used since before recorded history, and it is uncertain where they originated, the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site in south-eastern Iran, estimated to be from between 2800–2500 BCE. Other excavations from ancient tombs in the Indus Valley civilization indicate a South Asian origin, the Egyptian game of Senet was played with dice. Senet was played before 3000 BC and up to the 2nd century AD and it was likely a racing game, but there is no scholarly consensus on the rules of Senet. Dicing is mentioned as an Indian game in the Rigveda, Atharvaveda, there are several biblical references to casting lots, as in Psalm 22, indicating that dicing was commonplace when the psalm was composed. Knucklebones was a game of skill played by women and children, although gambling was illegal, many Romans were passionate gamblers who enjoyed dicing, which was known as aleam ludere. Dicing was even a popular pastime of emperors, letters by Augustus to Tacitus and his daughter recount his hobby of dicing. There were two sizes of Roman dice, tali were large dice inscribed with one, three, four, and six on four sides. Tesserae were smaller dice with sides numbered one to six. Twenty-sided dice date back to the 2nd century AD and from Ptolemaic Egypt as early as the 2nd century BC, dominoes and playing cards originated in China as developments from dice. The transition from dice to playing cards occurred in China around the Tang dynasty, in Japan, dice were used to play a popular game called sugoroku

17.
Role playing games
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A role-playing game is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, actions taken within many games succeed or fail according to a formal system of rules and guidelines. There are several forms of RPG, the original form, sometimes called the tabletop RPG, is conducted through discussion, whereas in live action role-playing games players physically perform their characters actions. In both of these forms, an arranger called a game master usually decides on the rules and setting to be used, acting as referee, while each of the other players plays the role of a single character. Several varieties of RPG also exist in media, such as multi-player text-based MUDs and their graphics-based successors. These games often share settings and rules with tabletop RPGs, despite this variety of forms, some game forms such as trading card games and wargames that are related to role-playing games may not be included. Role-playing activity may sometimes be present in games, but it is not the primary focus. The term is sometimes used to describe roleplay simulation games and exercises used in teaching, training. Both authors and major publishers of tabletop role-playing games consider them to be a form of interactive and collaborative storytelling, events, characters, and narrative structure give a sense of a narrative experience, and the game need not have a strongly-defined storyline. Interactivity is the difference between role-playing games and traditional fiction. Whereas a viewer of a show is a passive observer. Such role-playing games extend an older tradition of storytelling games where a party of friends collaborate to create a story. Participants in a game will generate specific characters and an ongoing plot. A consistent system of rules and a more or less realistic campaign setting in games aids suspension of disbelief, the level of realism in games ranges from just enough internal consistency to set up a believable story or credible challenge up to full-blown simulations of real-world processes. There is also a variety of systems of rules and game settings. Games that emphasize plot and character interaction over game mechanics and combat sometimes prefer the name storytelling game and these types of games tend to minimize or altogether eliminate the use of dice or other randomizing elements. Some games are played with characters created before the game by the GM and this type of game is typically played at gaming conventions, or in standalone games that do not form part of a campaign. Tabletop and pen-and-paper RPGs are conducted through discussion in a social gathering

18.
Enneagon
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In geometry, a nonagon /ˈnɒnəɡɒn/ is a nine-sided polygon or 9-gon. The name nonagon is a hybrid formation, from Latin, used equivalently, attested already in the 16th century in French nonogone. The name enneagon comes from Greek enneagonon, and is more correct. A regular nonagon is represented by Schläfli symbol and has angles of 140°. Although a regular nonagon is not constructible with compass and straightedge and it can be also constructed using neusis, or by allowing the use of an angle trisector. The following is a construction of a nonagon using a straightedge. The regular enneagon has Dih9 symmetry, order 18, there are 2 subgroup dihedral symmetries, Dih3 and Dih1, and 3 cyclic group symmetries, Z9, Z3, and Z1. These 6 symmetries can be seen in 6 distinct symmetries on the enneagon, john Conway labels these by a letter and group order. Full symmetry of the form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g9 subgroup has no degrees of freedom but can seen as directed edges. The regular enneagon can tessellate the euclidean tiling with gaps and these gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H with H representing *632 hexagonal symmetry in the plane, the K9 complete graph is often drawn as a regular enneagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex and they Might Be Giants have a song entitled Nonagon on their childrens album Here Come the 123s. It refers to both an attendee at a party at which everybody in the party is a many-sided polygon, slipknots logo is also a version of a nonagon, being a nine-pointed star made of three triangles. King Gizzard & the Lizard Wizard have an album titled Nonagon Infinity, temples of the Bahai Faith are required to be nonagonal. The U. S. Steel Tower is an irregular nonagon, enneagram Trisection of the angle 60°, Proximity construction Weisstein, Eric W. Nonagon

19.
Pyramid (geometry)
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base

20.
Monohedron
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In geometry a monogon is a polygon with one edge and one vertex. Since a monogon has only one side and only one vertex, in Euclidean geometry a monogon is a degenerate polygon because its endpoints must coincide, unlike any Euclidean line segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon, in spherical geometry, a monogon can be constructed as a vertex on a great circle. This forms a dihedron, with two hemispherical monogonal faces which share one 360° edge and one vertex and its dual, a hosohedron, has two antipodal vertices at the poles, one 360 degree lune face, and one edge between the two vertices. Digon Herbert Busemann, The geometry of geodesics, new York, Academic Press,1955 Coxeter, H. S. M, Regular Polytopes

21.
Dihedron
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A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons, a regular dihedron is the dihedron formed by two regular polygons, which may be described by the Schläfli symbol. As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere, the dual of a n-gonal dihedron is the n-gonal hosohedron, where n digon faces share two vertices. A dihedron can be considered a degenerate prism consisting of two n-sided polygons connected back-to-back, so that the object has no depth. The polygons must be congruent, but glued in such a way one is the mirror image of the other. This characterization holds also for the distances on the surface of a dihedron, as a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. The regular polyhedron is self-dual, and is both a hosohedron and a dihedron, in the limit the dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation, A regular ditope is an n-dimensional analogue of a dihedron, with Schläfli symbol. It has two facets, which share all ridges, in common, polyhedron Polytope Weisstein, Eric W. Dihedron

22.
Trihedron
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In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians, the restriction m ≥3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a tiling, this restriction may be relaxed, since digons can be represented as spherical lunes. Allowing m =2 admits a new class of regular polyhedra. On a spherical surface, the polyhedron is represented as n abutting lunes, all these lunes share two common vertices. The digonal faces of a 2n-hosohedron, represents the fundamental domains of symmetry in three dimensions, Cnv, order 2n. The reflection domains can be shown as alternately colored lunes as mirror images, bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n. The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the dual of the n-gonal hosohedron is the n-gonal dihedron. The polyhedron is self-dual, and is both a hosohedron and a dihedron, a hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism, in the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation, Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has two vertices, each with a vertex figure, the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M, Coxeter, and possibly derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”. Polyhedron Polytope McMullen, Peter, Schulte, Egon, Abstract Regular Polytopes, Cambridge University Press, ISBN 0-521-81496-0 Coxeter, H. S. M, ISBN 0-486-61480-8 Weisstein, Eric W. Hosohedron

23.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges

24.
Pentahedron
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In geometry, a pentahedron is a polyhedron with five faces. Since there are no face-transitive polyhedra with five sides and there are two distinct types, this term is less frequently used than tetrahedron or octahedron. With regular polygon faces, the two forms are the square pyramid and triangular prism. Geometric variations with irregular faces can also be constructed, the square pyramid can be seen as a degenerate triangular prism where one edge of its side edges is collapsed into a point, losing one edge and one vertex, and changing two squares into triangles. An irregular pentahedron can be a non-convex solid, there is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron, it exists as a spherical tiling of digon faces, called a pentagonal hosohedron with Schläfli symbol. It has 2 vertices,5 edges, and 5 digonal faces

25.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size

26.
Enneahedron
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In geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, the most familiar enneahedra are the octagonal pyramid and the heptagonal prism. The heptagonal prism is a polyhedron, with two regular heptagon faces and seven square faces. The octagonal pyramid has eight triangular faces around a regular octagonal base. Two more enneahedra are also found among the Johnson solids, the square pyramid. The three-dimensional associahedron, a near-miss Johnson solid with six pentagonal faces, five Johnson solids have enneahedral duals, the triangular cupola, gyroelongated square pyramid, self-dual elongated square pyramid, triaugmented triangular prism, and tridiminished icosahedron. Another enneahedron is the diminished trapezohedron with a base, and 4 kite and 4 triangle faces. The Herschel graph also represents the vertices and edges of an enneahedron and it is the simplest polyhedron without a Hamiltonian cycle, the only enneahedron in which all faces have the same number of edges, and one of only three bipartite enneahedra. The two smallest isospectral polyhedral graphs are enneahedra with eight vertices each, like the rhombic dodecahedron itself, this shape can be used to tessellate three-dimensional space. An elongated form of shape that still tiles space can be seen atop the rear side towers of the 12th-century Romanesque Basilica of Our Lady. The towers themselves, with their four pentagonal sides, four roof facets, more generally, Goldberg found at least 40 topologically distinct space-filling enneahedra. There are 2606 topologically distinct convex enneahedra, excluding mirror images and these can be divided into subsets of 8,74,296,633,768,558,219,50, with 7 to 14 vertices respectively. A table of numbers, together with a detailed description of the nine-vertex enneahedra, was first published in the 1870s by Thomas Kirkman. Enumeration of Polyhedra by Steven Dutch Weisstein, Eric W. Nonahedron

27.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does

28.
Tetradecahedron
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A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces, a tetradecahedron is sometimes called a tetrakaidecahedron. No difference in meaning is ascribed, the Greek word kai means and. There is evidence that mammalian cells are shaped like flattened tetrakaidecahedra. There are 1,496,225,352 topologically distinct convex tetradecahedra, excluding mirror images, with Greek Numerical Prefixes Weisstein, Eric W. Tetradecahedron

29.
Octadecahedron
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In geometry, an octadecahedron is a polyhedron with 18 faces. No octadecahedron is regular, hence, the name does not commonly refer to one specific polyhedron, in chemistry, the octadecahedron commonly refers to a specific structure with C2v symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion 2−, there are 107,854,282,197,058 topologically distinct convex octadecahedra, excluding mirror images, having at least 11 vertices. The most familiar octadecahedra are the pyramid, hexadecagonal prism. The hexadecagonal prism and the octagonal antiprism are uniform polyhedra, with regular bases, four more octadecahedra are also found among the Johnson solids, the square gyrobicupola, the square orthobicupola, the elongated square cupola, and the sphenomegacorona. In addition, some uniform polyhedra are also octadecahedra

30.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio

31.
Rhombic triacontahedron
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In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types and it is a Catalan solid, and the dual polyhedron of the icosidodecahedron. The ratio of the diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1 = tan−1. A rhombus so obtained is called a golden rhombus, being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids and it contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra, the plane of each face is perpendicular to the center of the rhombic triacontahedron, and is located at the same distance. Using one of the three golden rectangles drawn into the inscribed icosahedron we can easily deduce the distance between the center of the solid and the center of its rhombic face. The rhombic triacontahedron can be dissected into 20 golden rhombohedra,10 acute ones and 10 flat ones, danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron, the simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube. Roger von Oechs Ball of Whacks comes in the shape of a rhombic triacontahedron, the rhombic triacontahedron is used as the d30 thirty-sided die, sometimes useful in some roleplaying games or other places. The rhombic triacontahedron has three positions, two centered on vertices, and one mid-edge. Embedded in projection 10 are the fat rhombus and skinny rhombus which tile together to produce the non-periodic tessellation often referred to as Penrose tiling, the rhombic triacontahedron has over 227 stellations. This polyhedron is a part of a sequence of rhombic polyhedra, the cube can be seen as a rhombic hexahedron where the rhombi are also rectangles. The rhombic triacontahedron forms the hull of one projection of a 6-cube to 3 dimensions. Truncated rhombic triacontahedron Rhombille tiling Golden rhombus Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design

32.
Rhombic enneacontahedron
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A rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim, the rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron. This construction is expressed in the Conway polyhedron notation jtI with join operator j, without the equal edge constraint, the wide rhombi are kites if limited only by the icosahedral symmetry. The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, the face angles of these rhombi are approximately 70. 528° and 109. 471°. The thirty slim rhombic faces have face vertex angles of 41. 810° and 138. 189° and it is also called a rhombic enenicontahedron in Lloyd Kahns Domebook 2. The optimal packing fraction of rhombic enneacontahedra is given by η =16 −345 ≈0.7947377530014315 and it was noticed that this optimal value is obtained in a Bravais lattice by de Graaf. VRML model, George Hart, George Harts Conway Generator Try dakD Domebook2 by Kahn, Lloyd, Easton, Bob, Calthorpe, Peter, et al. Pacific Domes, Los Gatos, CA, page 102 de Graaf, J. van Roij, R. Dijkstra, M. Dense Regular Packings of Irregular Nonconvex Particles, Phys. 107,155501, arXiv,1107.0603, Bibcode, 2011PhRvL. 107o5501D, doi,10. 1103/PhysRevLett.107.155501 Torquato, S. Jiao, Y. Dense packings of the Platonic and Archimedean solids, Nature,460,876, arXiv,0908.4107, Bibcode, 2009Natur.460. 876T, doi,10. 1038/nature08239, PMID19675649 Hales, Thomas C. A proof of the Kepler conjecture, Annals of Mathematics,162,1065, doi,10. 4007/annals.2005.162.1065 Weisstein, Eric W. Rhombic enneacontahedron

33.
Apeirohedron
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Skew apeirohedra have also been called polyhedral sponges. Many are directly related to a uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. Characteristically, an infinite skew polyhedron divides 3-dimensional space into two halves, if one half is thought of as solid the figure is sometimes called a partial honeycomb. According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular polygons to regular skew polyhedra. Coxeter and Petrie found three of these that filled 3-space, There also exist chiral skew apeirohedra of types, and these skew apeirohedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror symmetric. Beyond Euclidean 3-space, in 1967 C. W. L. Garner published a set of 31 regular skew polyhedra in hyperbolic 3-space. J. Richard Gott in 1967 published a set of seven infinite skew polyhedra which he called regular pseudopolyhedrons. Gott relaxed the definition of regularity to allow his new figures, where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent. Thus, Gotts new examples are not regular by Coxeter and Petries definition, however neither the term pseudopolyhedron nor Gotts definition of regularity have achieved wide usage. Wells in 1960s also published a list of skew apeirohedra, There are two prismatic forms,5 squares on a vertex,8 triangles on a vertex is also formed from parallel planes of triangular tilings, with alternating octahedral holes going both ways. Is composed of 3 coplanar pentagons around a vertex and two perpendicular pentagons filling the gap, Gott also acknowledged that there are other periodic forms of the regular planar tessellations. Both the square tiling and triangular tiling can be curved into approximating infinite cylinders in 3-space and he wrote some theorems, For every regular polyhedron, *<4. The number of surrounding a given face is p* in any regular generalized polyhedron. Every regular pseudopolyhedron approximates a curved surface. The seven regular pseudopolyhedron are repeating structures, There are many other uniform skew apeirohedra. Wachmann, Burt and Kleinmann discovered many examples but it is not known whether their list is complete and they can be named by their vertex configuration, although it is not a unique designation for skew forms. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, The Regular Sponges, or Skew Polyhedra, Scripta Mathematica 6 240-244, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, ISBN 978-1-56881-220-5 Schulte, Egon, Chiral polyhedra in ordinary space