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.
Tetradecagonal 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

5.
Heptagonal antiprism
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In geometry, the heptagonal antiprism is the fifth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to other. In the case of a regular 7-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, as faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles. If faces are all regular, it is a semiregular polyhedron, virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A7

6.
Tetrahedral symmetry
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A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4, chiral and full are discrete point symmetries. They are among the point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles in the plane, each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these meet at order 2 and 3 gyration points. T,332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry, there are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A4, the group on 4 elements, in fact it is the group of even permutations of the four 3-fold axes. The three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. Td, *332, or 43m, of order 24 – achiral or full tetrahedral symmetry and this group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 axes, td and O are isomorphic as abstract groups, they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion, see also the isometries of the regular tetrahedron. This group has the same axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 axes, and there is an inversion symmetry. Th is isomorphic to T × Z2, every element of Th is either an element of T, apart from these two normal subgroups, there is also a normal subgroup D2h, of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the product of the normal subgroup of T with Ci. The quotient group is the same as above, of type Z3, the three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. It is the symmetry of a cube with on each face a line segment dividing the face into two rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the permutations of the body diagonals

7.
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

8.
Tetrahedrally diminished dodecahedron
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In geometry, a tetrahedrally diminished dodecahedron is a topologically self-dual polyhedron made of 16 vertices,30 edges, and 16 faces. Within its tetrahedral symmetry, it has geometric varied proportions, by Dorman Luke dual construction, a unique geometric proportion can be defined. The kite faces have edges of length ratio ~1,0.6325, as a self-dual hexadecahedron, it is one of 302404 forms,1476 with at least order 2 symmetry, and the only one with tetrahedral symmetry. As a diminished regular dodecahedron, with 4 vertices removed, the faces are trapezoids. As a stellation of the regular icosahedron it is one of 32 stellations defined with tetrahedral symmetry, in Conway polyhedron notation, it can represented as pT, applying George W. Harts propeller operator to a regular tetrahedron

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