"the finite subgroups of SO(3), are: the cyclic groups ""C"" (the rotation group of a canonical pyramid), the dihedral groups ""D"" (the rotation group of a uniform prism, or canonical bipyramid), and the rotation groups ""T"", ""O"" and ""I"" of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron. In particular, the dihedral groups ""D"", ""D"" etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore, it is also called a ""dihedron"" (Greek: solid with two faces), which explains the name ""dihedral group"". The rotation group of"
"is a face with dimension 1. A flat shape in 3D space, like a triangle, is called a face with dimension 2 – or a 'facet'. A shape in 3D space, like a cube, is called a face with dimension 3 – or a 'volume'. The Computational Geometry Algorithms Library, or CGAL, represents Nef Polyhedra by using two main data structures. The first is a 'Sphere map' and the second is a 'Selective Nef Complex' (or SNC). The 'sphere map' stores information about the polyhedron by creating an imaginary sphere around each vertex, and painting it with various points and"
"Hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and because its faces cannot be equilateral triangles. It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The hexagonal"
"12 triangles and 2 triamonds . It may also be described as having a hybrid square-pentagonal antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then augmented with a pyramid. The dissected regular icosahedron is a name for this polytope with the two sets of 3 coplanar faces as trapezoids. This is the vertex figure of a 4D polytope, grand antiprism. It has 10 vertices, 23 edges, and 11 equilateral triangular faces and 2 trapezoid faces. In chemistry, this polyhedron is most commonly called the octadecahedron, for 18 triangular faces, and represents the"
"with order 2,3,n mirrors at each triangle face vertex. Hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and because its faces cannot be equilateral triangles. It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with"
"Snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron or dodecadeltahedron is a three-dimensional convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces) and one of the 92 Johnson solids (non-uniform convex polyhedra with regular faces). This shape was called a ""Siamese dodecahedron"" in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra. The ""dodecadeltahedron"""
"""E"" is the number of edges, and ""F"" is the number of faces. This equation is known as Euler's 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, and hence 6 faces. 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"
"name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no"
"Deltahedron In geometry, a deltahedron (plural ""deltahedra"") is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta (formula_1), which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra. There are only eight strictly-convex deltahedra: three are regular polyhedra, and five are Johnson solids. In the 6-faced deltahedron, some vertices have degree 3 and some"
"Hemi-cuboctahedron A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron. It has 4 triangular faces and 3 square faces, 12 edges, and 6 vertices. It can be seen as a rectified hemi-octahedron or rectified hemi-cube. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles and 3 square), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected. Its dual polyhedron is a rhombic hemi-dodecahedron which has 7 vertices (1-7), 12 edges (a-l), and 6 rhombic faces"
"is a subset of 12's proper divisors that add up to 12 (all of them but with 4 excluded), 12 is a semiperfect number. If an odd perfect number is of the form 12""k"" + 1, it has at least twelve distinct prime factors. A twelve-sided polygon is a dodecagon. A twelve-faced polyhedron is a dodecahedron. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. Twelve is a pentagonal number. The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for ""any"" arrangement of spheres (the Kepler"
"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 often referred to as such. A ""regular icosahedron"" can be distorted or marked up as a lower pyritohedral symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudo-icosahedron. This can be seen as an alternated truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle"
"Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces"
"Face (geometry) In solid geometry, a face is a flat (planar) 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, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). 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, and tile of a Euclidean plane tessellation. For"
"Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), 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 and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point."
"Truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicuboctahedron. The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all"
"with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles. Conway calls it a kisdeltille, constructed as a kis operation applied to a triangular tiling (deltille). In Japan the pattern is called asanoha for ""hemp leaf"", although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron. It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at"
"Trigonal trapezohedron In geometry, a trigonal trapezohedron or trigonal deltohedron is a three-dimensional figure formed by six congruent rhombi. Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The ""acute"" or ""prolate"" form has three acute angles corners of the rhombic faces meeting at two polar axis vertices. The ""obtuse"" or ""oblate"" or ""flat"" form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices. A trigonal trapezohedron is a special case of a rhombohedron. A general rhombohedron allows up to three types of rhombic faces. A trigonal trapezohedron is a special"
"(A-F). It has a real presentation as a uniform star polyhedron, the tetrahemihexahedron. Hemi-cuboctahedron A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron. It has 4 triangular faces and 3 square faces, 12 edges, and 6 vertices. It can be seen as a rectified hemi-octahedron or rectified hemi-cube. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles and 3 square), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected. Its dual polyhedron is a rhombic"
"Dodecahedron In geometry, a dodecahedron (Greek , from ""dōdeka"" ""twelve"" + ""hédra"" ""base"", ""seat"" or ""face"") is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular 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, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the"
"of five planes in general position in Euclidean space, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration . This construction is closely related to the property that every projective plane that can be embedded into a projective space obeys Desargues' theorem. This three-dimensional realization of the Desargues configuration is also called the complete pentahedron . The 5-cell or pentatope (a regular simplex in four dimensions) has five vertices, ten edges, ten triangular ridges (2-dimensional faces), and five tetrahedral facets; the edges"
"For ""p"" > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has eight faces, it is an octahedron. However, the term ""octahedron"" is primarily used to refer to the ""regular octahedron"", which has eight triangular faces. Because of the ambiguity of the term ""octahedron"" and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long"
"connected to each other by three edges, define a triangle, which is the simplest polygon in Euclidean space. More complex polygons can be created out of multiple triangles, or as a single object with more than 3 vertices. Four sided polygons (generally referred to as quads) and triangles are the most common shapes used in polygonal modeling. A group of polygons, connected to each other by shared vertices, is generally referred to as an element. Each of the polygons making up an element is called a face. In Euclidean geometry, any three non-collinear points determine a plane. For this reason,"
"Tetradecahedron 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 epidermis cells are shaped like flattened tetrakaidecahedra, an idea first suggested by Lord Kelvin. There are 1,496,225,352 topologically distinct ""convex"" tetradecahedra, excluding mirror images, having at least 9 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to"
"Snub dodecahedron In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror images (or ""enantiomorphs"") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron. Kepler"
"doubles the number of dimensions. A polytope constructed in such a unitary space is called a complex polytope. Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are ""hemi-icosahedra"" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the ""same"" face (Grünbaum, 1977). The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12. This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An"
"Enneahedron In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular. The most familiar enneahedra are the octagonal pyramid and the heptagonal prism. The heptagonal prism is a uniform polyhedron, with two regular heptagon faces and seven square faces. The octagonal pyramid has eight isosceles triangular faces around a regular octagonal base. Two more enneahedra are also found among the Johnson solids: the elongated square pyramid and the elongated triangular bipyramid. The three-dimensional associahedron, a"
"Euclidean 3-space. John Horton Conway calls it a ""pyramidille"". Cells can be seen in a translational cube, using 4 vertices on one face, and the cube center. Edges are colored by how many cells are around each of them. It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells. There are two types of planes of faces: one as a square tiling, and flattened triangular tiling with half of the triangles removed as ""holes"". It is dual to the truncated cubic honeycomb with octahedral and truncated cubic cells: If"
"Heptahedron A heptahedron (plural: heptahedra) is a polyhedron having seven sides, or faces. A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven faces form a rudimentary projective plane. No heptahedra are regular. There are 34 topologically distinct ""convex"" heptahedra, excluding mirror images. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles"
"example, any of the six squares that bound a cube is a face of the cube. Sometimes ""face"" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets (flat polygons formed by coplanar vertices which do not lie in the same face of the polyhedron). Any convex polyhedron's surface has Euler characteristic where ""V"" is the number of vertices,"
"Hessian polyhedron In geometry, the Hessian polyhedron is a regular complex polyhedron {3}{3}, , in formula_1. It has 27 vertices, 72 {} edges, and 27 {3} faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the ""Hessian configuration"" formula_2 or (912), 9 points lying by threes on twelve lines, with four lines through each point. Its complex reflection group is [3][3] or , order 648, also called a Hessian group. It has 27 copies of , order 24, at each vertex. It has 24 order-3 reflections. Its Coxeter number is 12, with degrees of the fundamental"
"a simple polygon is called a mouth if the diagonal lies outside the boundary of . Any convex polyhedron's surface has Euler characteristic where is the number of vertices, is the number of edges, and is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 12 edges and 6 faces, and hence 8 vertices. In computer graphics, objects are often represented as triangulated polyhedra in which the object vertices are associated"
"Triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four. The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, (""J""). As a Johnson solid with"