1.
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

2.
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

3.
Regular polyhedron
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A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive, in classical contexts, many different equivalent definitions are used, a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form, there are 5 finite convex regular polyhedra, known as the Platonic solids. These are the, tetrahedron, cube, octahedron, dodecahedron and icosahedron, there are also four regular star polyhedra, making nine regular polyhedra in all. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons, All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre, An insphere, tangent to all faces, an intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices, the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them, Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry, the five Platonic solids have an Euler characteristic of 2. Some of the stars have a different value. The sum of the distances from any point in the interior of a polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra, in a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows, The tetrahedron is self-dual, the cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other, the small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other, the Schläfli symbol of the dual is just the original written backwards, for example the dual of is. See also Regular polytope, History of discovery, stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery, the earliest known written records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, euclids reference to Plato led to their common description as the Platonic solids

4.
Hexadecagonal pyramid
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base

5.
Pentadecagonal prism
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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

6.
Laves graph
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In geometry and crystallography, the Laves graph is an infinite cubic symmetric graph. It can be embedded into space, with integer coordinates. It can also be defined abstractly as a covering graph of the complete graph on four vertices. H. S. M. Coxeter named this graph after Fritz Laves and it has also been called the K4 crystal, -a network, diamond twin, triamond, and the srs net. As Coxeter describes, the vertices of the Laves graph can be defined by selecting one out of eight points in the three-dimensional integer lattice. Specifically, one chooses the points, and all the points that can be formed by adding multiples of four to these coordinates. The edges of the Laves graph connect pairs of points whose Euclidean distance from each other is √2, the other non-adjacent pairs of vertices are farther apart, at a distance of at least √6 from each other. The two copies are mirror images of each other, as an abstract graph, the Laves graph can be constructed as the maximal abelian covering graph of the complete graph K4. That is, the graph whose vertices are orbits and whose edges are adjacent pairs of orbits is exactly K4 and this construction justifies one of the alternative names of the Laves graph, the K4 crystal. Then, fix the set of vertices of the graph to be the ordered pairs where v is a vertex of G and w is a vector in Z d. The resulting graph is independent of the choice of the tree. The d-dimensional integer lattice is the maximal abelian covering graph of a graph with one vertex, the Laves graph is a cubic graph and a symmetric graph. The girth of this structure is 10 — the shortest cycles in the graph have 10 vertices — and 15 of these pass through each vertex. The cells of the Voronoi diagram of this structure are heptadecahedra with 17 faces each and they are plesiohedra, polyhedra that tile space isohedrally. Experimenting with the structures formed by these polyhedra led Alan Schoen to discover the gyroid minimal surface, computations suggest that the Laves graph can serve as the pattern for a metastable or perhaps unstable allotrope of carbon. 5°. The Laves graph may also give a structure for boron. Other chemicals that may form this structure include SrSi2, and elemental nitrogen, the structure of the Laves graph, and of gyroid surfaces derived from it, has also been observed experimentally in soap-water systems, and in the chitin networks of butterfly wing scales. Baez, John, Laves Graph, Visual Insight, American Mathematical Society

7.
Voronoi diagram
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In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points is specified beforehand, and for each seed there is a region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells, the Voronoi diagram of a set of points is dual to its Delaunay triangulation. It is named after Georgy Voronoi, and is called a Voronoi tessellation, a Voronoi decomposition. Voronoi diagrams have practical and theoretical applications to a number of fields, mainly in science and technology. They are also known as Thiessen polygons, in the simplest case, shown in the first picture, we are given a finite set of points in the Euclidean plane. Each such cell is obtained from the intersection of half-spaces, the line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices are the points equidistant to three sites, let X be a metric space with distance function d. Let K be a set of indices and let k ∈ K be a tuple of nonempty subsets in the space X. In other words, if d = inf denotes the distance between the point x and the subset A, then R k = The Voronoi diagram is simply the tuple of cells k ∈ K. In principle some of the sites can intersect and even coincide, in addition, infinitely many sites are allowed in the definition, but again, in many cases only finitely many sites are considered. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram, however, in general the Voronoi cells may not be convex or even connected. In the usual Euclidean space, we can rewrite the definition in usual terms. Each Voronoi polygon R k is associated with a generator point P k, let X be the set of all points in the Euclidean space. Let P1 be a point that generates its Voronoi region R1, P2 that generates R2, and P3 that generates R3, and so on. Then, as expressed by Tran et al all locations in the Voronoi polygon are closer to the point of that polygon than any other generator point in the Voronoi diagram in Euclidian plane. As a simple illustration, consider a group of shops in a city, suppose we want to estimate the number of customers of a given shop. With all else being equal, it is reasonable to assume that customers choose their preferred shop simply by distance considerations, they will go to the shop located nearest to them

8.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules

9.
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

10.
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

11.
Hosohedron
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In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians, the restriction m ≥3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a tiling, this restriction may be relaxed, since digons can be represented as spherical lunes. Allowing m =2 admits a new class of regular polyhedra. On a spherical surface, the polyhedron is represented as n abutting lunes, all these lunes share two common vertices. The digonal faces of a 2n-hosohedron, represents the fundamental domains of symmetry in three dimensions, Cnv, order 2n. The reflection domains can be shown as alternately colored lunes as mirror images, bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n. The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the dual of the n-gonal hosohedron is the n-gonal dihedron. The polyhedron is self-dual, and is both a hosohedron and a dihedron, a hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism, in the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation, Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has two vertices, each with a vertex figure, the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M, Coxeter, and possibly derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”. Polyhedron Polytope McMullen, Peter, Schulte, Egon, Abstract Regular Polytopes, Cambridge University Press, ISBN 0-521-81496-0 Coxeter, H. S. M, ISBN 0-486-61480-8 Weisstein, Eric W. Hosohedron

12.
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

13.
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

14.
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

15.
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

16.
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

17.
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

18.
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

19.
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

20.
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

21.
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

22.
Skew 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