"Great triambic icosahedron In geometry, the great triambic icosahedron and medial triambic icosahedron are visually identical dual uniform polyhedra. The exterior surface also represents the Def stellation of the icosahedron. The only way to differentiate these two polyhedra is to mark which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres. The 12 vertices of the convex hull matches the vertex arrangement of an icosahedron. The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron, U47. It has 20 inverted-hexagonal faces, shaped like a three-bladed propeller."
"Hexadecahedron A hexadecahedron (or hexakaidecahedron) is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism. There are 387,591,510,244 topologically distinct ""convex"" hexadecahedra, excluding mirror images, having at least 10 vertices. (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 between edges or faces.) There are 302,404 self-dual hexadecahedron, 1476"
"Heptadecahedron A heptadecahedron (or heptakaidecahedron) is a polyhedron with 17 faces. No heptadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a heptadecahedron, for example the hexadecagonal pyramid and pentadecagonal prism. There are 6,415,851,530,241 topologically distinct ""convex"" heptadecahedra, excluding mirror images, having at least 11 vertices. (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 between edges or faces.) The infinite Laves graph has convex heptadecahedral Voronoi"
"Enneadecahedron A enneadecahedron (or enneakaidecahedron) is a polyhedron with 19 faces. No enneadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of an enneadecahedron, for example the octadecagonal (18-sided) pyramid, or the heptadecagonal (17-sided) prism; the latter is the only convex enneadecahedron with all regular polygonal faces. There are many topologically distinct ""convex"" enneadecahedra, excluding mirror images, having at least 12 vertices. (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"
"Pentakis icosidodecahedron The pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the ""truncated rhombic triacontahedron"" (chamfered dodecahedron). Its name comes from a topological construction from the icosidodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained even with the 12 order-5 vertices at a different distance from the center as the other 30. It can also be topologically constructed from the"
"Octadecahedron In geometry, an octadecahedron (or octakaidecahedron) 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 C symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [BH]. There are 107,854,282,197,058 topologically distinct ""convex"" octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one"
"Hendecahedron A hendecahedron (or undecahedron) is a polyhedron with 11 faces. There are numerous topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism. Three forms are Johnson solids: augmented hexagonal prism, biaugmented triangular prism, and elongated pentagonal pyramid. Two classes, the bisymmetric and the sphenoid hendecahedra, are space-filling. The name of hendecahedron is based on its meaning. Hen- represents one. Deca represents ten, and then combined with the polyhedron suf-hedron, it is Hendecahedron. In all the convex hendecahedrons, there are a total of 440,564 convex ones with distinct differences in topology. There are significant differences"
"or the angles between edges or faces.) Enneadecahedron A enneadecahedron (or enneakaidecahedron) is a polyhedron with 19 faces. No enneadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of an enneadecahedron, for example the octadecagonal (18-sided) pyramid, or the heptadecagonal (17-sided) prism; the latter is the only convex enneadecahedron with all regular polygonal faces. There are many topologically distinct ""convex"" enneadecahedra, excluding mirror images, having at least 12 vertices. (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"
"Icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron. An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron located at the midpoints of the edges of either. Its dual polyhedron is the rhombic triacontahedron."
"icosahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices. From this construction, all 80 triangles will be equilateral, but faces will be coplanar. It represents the exterior envelope of a vertex-centered orthogonal projection of the 600-cell, one of six convex regular 4-polytopes, into 3 dimensions. Pentakis icosidodecahedron The pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the ""truncated rhombic triacontahedron"" (chamfered dodecahedron). Its name comes from a topological construction from the icosidodecahedron with the kis operator applied to the pentagonal faces. In"
"Expanded icosidodecahedron The expanded icosidodecahedron is a polyhedron, constructed as a expanded icosidodecahedron. It has 122 faces: 20 triangles, 60 squares, 12 pentagons, and 30 rhombs. The 120 vertices exist at two sets of 60, with a slightly different distance from its center. It can also be constructed as a rectified rhombicosidodecahedron. The expansion operation from the rhombic triacontahedron can be seen in this animation: This polyhedron can be dissected into a central rhombic triacontahedron surrounded by: 30 rhombic prisms, 20 tetrahedra, 12 pentagonal pyramids, 60 triangular prisms. If the central rhombic triacontahedron and the 30 rhombic prisms are removed,"
"cells. Because of the symmetries of the graph, these heptadecahedra are plesiohedra form an isohedral tessellation of three-dimensional space. Heptadecahedron A heptadecahedron (or heptakaidecahedron) is a polyhedron with 17 faces. No heptadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a heptadecahedron, for example the hexadecagonal pyramid and pentadecagonal prism. There are 6,415,851,530,241 topologically distinct ""convex"" heptadecahedra, excluding mirror images, having at least 11 vertices. (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"
"dodecahedron, with 92 faces, in second place. If the bipyramids and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other strictly convex polyhedron where every face of the polyhedron has the same shape. Projected into a sphere, the edges of a disdyakis triacontahedron define 15 great circles. Buckminster Fuller used these 15 great circles, along with 10 and 6 others in two other polyhedra to define his 31 great circles of the spherical icosahedron. The edges of the polyhedron projected onto a sphere form 15 great circles, and represent all 15 mirror planes of reflective"
"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"
"resembles an icosahedron, because the great dodecahedron is completely contained inside the icosahedron. Small complex icosidodecahedron In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron. A small complex icosidodecahedron can be constructed from a number of different vertex figures. The small complex icosidodecahedron can be seen as a compound of the icosahedron {3,5} and the great dodecahedron"
"(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 between edges or faces.) Hendecahedron A hendecahedron (or undecahedron) is a polyhedron with 11 faces. There are numerous topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism. Three forms are Johnson solids: augmented hexagonal prism, biaugmented triangular prism, and elongated pentagonal pyramid. Two classes, the bisymmetric and the sphenoid hendecahedra, are space-filling. The name of hendecahedron is based on"
"Edge-contracted icosahedron In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices with C symmetry, order 4. It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 faces. This contraction distorts the circumscribed sphere original vertices. With all equilateral triangle faces, it has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid. If the sets of coplanar triangles are considered a single face (called a triamond), it has 10 vertices, 22 edges, and 14 faces,"
"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"
"Truncated icosidodecahedron In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has more vertices (120) and edges (180) than any other convex nonprismatic uniform polyhedron. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron. The name ""great rhombicosidodecahedron"" refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).<br> There is a nonconvex uniform polyhedron with a similar name, the"
"hebesphenorotunda. In addition, some uniform star polyhedra are also octadecahedra: Octadecahedron In geometry, an octadecahedron (or octakaidecahedron) 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 C symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [BH]. There are 107,854,282,197,058 topologically distinct ""convex"" octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces"
"the icosahedron itself. One is a regular Kepler–Poinsot polyhedron. Three are regular compound polyhedra. The small stellated dodecahedron, great dodecahedron, and great icosahedron are three facetings of the regular icosahedron. They share the same vertex arrangement. They all have 30 edges. The regular icosahedron and great dodecahedron share the same edge arrangement but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles). There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to"
"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"
"Small complex icosidodecahedron In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron. A small complex icosidodecahedron can be constructed from a number of different vertex figures. The small complex icosidodecahedron can be seen as a compound of the icosahedron {3,5} and the great dodecahedron {5,5/2} where all vertices are precise and edges coincide. The small complex icosidodecahedron"
"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"
"with at least order 2 symmetry. The high symmetry self-dual has chiral tetrahedral symmetry, and can be seen topologically by removing 4 of 20 vertices of a regular dodecahedron and is called a tetrahedrally diminished dodecahedron. Hexadecahedron A hexadecahedron (or hexakaidecahedron) is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism. There are 387,591,510,244 topologically distinct ""convex"" hexadecahedra, excluding mirror images, having at least 10 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different"
"with 15 and 10 others in two other polyhedra to define his 31 great circles of the spherical icosahedron. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of: where ""φ"" is the golden ratio, . The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A and H Coxeter planes. The surface area ""A"" and the volume ""V"" of the icosidodecahedron of edge length ""a"" are: The icosidodecahedron can also be represented as a"
"you can create a toroidal polyhedron with all regular polygon faces. Expanded icosidodecahedron The expanded icosidodecahedron is a polyhedron, constructed as a expanded icosidodecahedron. It has 122 faces: 20 triangles, 60 squares, 12 pentagons, and 30 rhombs. The 120 vertices exist at two sets of 60, with a slightly different distance from its center. It can also be constructed as a rectified rhombicosidodecahedron. The expansion operation from the rhombic triacontahedron can be seen in this animation: This polyhedron can be dissected into a central rhombic triacontahedron surrounded by: 30 rhombic prisms, 20 tetrahedra, 12 pentagonal pyramids, 60 triangular prisms. If"
"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"
"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"
"dodecahedron forms an icosidodecahedron. The regular dodecahedron has icosahedral symmetry I, Coxeter group [5,3], order 120, with an abstract group structure of ""A"" × ""Z"". When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...), which ratio is approximately , or in exact terms: or . A regular dodecahedron has 12 faces and 20 vertices,"
"Great complex icosidodecahedron In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as topological polyhedron. It can be constructed from a number of different vertex figures. The great complex icosidodecahedron can be considered a compound of the small stellated dodecahedron, {5/2,5}, and great icosahedron, {3,5/2}, sharing the same vertices and edges, while the second is hidden, being completely contained inside the first. Its two-dimensional"
"Rhombicosidodecahedron In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices, and 120 edges. Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a ""rhombicosidodecahedron"", being short for ""truncated icosidodecahedral rhombus"", with ""icosidodecahedral rhombus"" being his name for a rhombic triacontahedron. There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the"
"Excavated dodecahedron In geometry, the excavated dodecahedron is a star polyhedron having 60 equilateral triangular faces. Its exterior surface represents the Efg stellation of the icosahedron. It appears in Magnus Wenninger's book ""Polyhedron Models"" as model 28, the ""third stellation of icosahedron"". It has the same external form as a certain facetting of the dodecahedron having 20 self-intersecting hexagons as faces. The non-convex hexagon face can be broken up into four equilateral triangles, three of which are the same size. A true excavated dodecahedron has the three congruent equilateral triangles as true faces of the polyhedron, while the interior equilateral"