How many faces has a dodecahedron?
"Regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}. If the edge length of a regular dodecahedron is ""a"", the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's"
"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"
"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"
"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"
"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"
"even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,""n"" mirrors at each triangle face vertex. Disdyakis dodecahedron In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It superficially resembles an inflated rhombic dodecahedron—if one replaces"
"(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"
"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"
"Disdyakis dodecahedron In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It superficially resembles an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. It is the net of a rhombic dodecahedral pyramid. It has O octahedral"
"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"
"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"
"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"
"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"
"tritruncated rhombic dodecahedron, a truncation of the order-3 vertices of the rhombic dodecahedron. The 8 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles. The hexagonal faces are equilateral but not regular. The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the"
"prisms. If the central rhombic dodecahedron and the 12 rhombic prisms are removed, you can create a toroidal polyhedron with all regular polygon faces. This toroid has 86 faces (8 triangles and 78 squares), 168 edges, and 62 vertices. 14 of the 62 vertices are on the interior, defining the removed central rhombic dodecahedron. With Euler characteristic χ = f + v - e = -20, its genus, g = (2-χ)/2 is 11. Expanded cuboctahedron The expanded cuboctahedron is a polyhedron, constructed as a expanded cuboctahedron. It has 50 faces: 8 triangles, 30 squares, and 12 rhombs. The 48 vertices"
"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"""
"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"
"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"
"Truncated dodecadodecahedron In geometry, the truncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U. It is given a Schläfli symbol t{5/3,5}. It has 120 vertices and 54 faces: 30 squares, 12 decagons, and 12 decagrams. The central region of the polyhedron is connected to the exterior via 20 small triangular holes. The name ""truncated dodecadodecahedron"" is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by . For this reason,"
"Dodecahedral prism In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms. It has 54 faces: 30 squares and 24 pentagons. It has 80 edges and 40 vertices. It can be constructed by creating two coinciding dodecahedra in 3-space, and translating each copy in opposite perpendicular directions in 4-space until their separation equals their edge length. It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids or Archimedean solids. The dodecahedral prism consists of two dodecahedra"
"Pentakis snub dodecahedron The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry. Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles. The path between the valence-5 vertices is two edges in a row, and then"
"Icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . 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 nonconvex, 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. Both have icosahedral symmetry. The term ""regular icosahedron"" generally refers to the convex variety, while the nonconvex"
"with the small dodecahemicosahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the pentagonal faces in common). This polyhedron can be considered a rectified great dodecahedron. It is center of a truncation sequence between a small stellated dodecahedron and great dodecahedron: The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). The truncation of the dodecadodecahedron itself is not uniform and attempting to make it uniform results in a degenerate polyhedron (that looks like a small rhombidodecahedron"
"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"
"Diminished rhombic dodecahedron In geometry, a diminished rhombic dodecahedron is a rhombic dodecahedron with one or more vertices removed. This article describes diminishing one 4-valence vertex. This diminishment creates one new square face while 4 rhombic faces are reduced to triangles. It has 13 vertices, 24 edges, and 13 faces. It has ""C"" symmetry, order 8. Like the rhombic dodecahedron, the long diagonal of each rhombic face is times the length of the short diagonal, so that the acute angles on each face measure arccos(), or approximately 70.53°. Like the dihedral symmetry pyramids, and elongated pyramids, it is self-dual, with"
"Elongated dodecahedron In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism. Along with the rhombic dodecahedron, it is a space-filling polyhedron. This is related to the rhombic dodecahedral honeycomb with an elongation of zero. Projected normal to the elongation direction, the honeycomb looks like a square tiling with the rhombi projected into squares. The expanded"
"a turn and one more edge. Pentakis snub dodecahedron The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry. Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles. The path between the valence-5 vertices is two"
"dodecahedral prism into 3D space has a dodecahedral envelope. The two dodecahedral cells project onto the entire volume of this envelope, while the 12 decagonal prismic cells project onto its 12 pentagonal faces. Dodecahedral prism In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms. It has 54 faces: 30 squares and 24 pentagons. It has 80 edges and 40 vertices. It can be constructed by creating two coinciding dodecahedra in 3-space, and translating each copy in opposite perpendicular directions in 4-space until their separation equals their"
"Expanded cuboctahedron The expanded cuboctahedron is a polyhedron, constructed as a expanded cuboctahedron. It has 50 faces: 8 triangles, 30 squares, and 12 rhombs. The 48 vertices exist at two sets of 24, with a slightly different distance from its center. It can also be constructed as a rectified rhombicuboctahedron. The expansion operation from the rhombic dodecahedron can be seen in this animation: The ""expanded cuboctahedron"" can fill space along with a cuboctahedron, octahedron, and triangular prism. This polyhedron can be dissected into a central rhombic dodecahedron surrounded by: 12 rhombic prisms, 8 tetrahedra, 6 square pyramids, and 24 triangular"
"Small complex rhombicosidodecahedron In geometry, the small complex rhombicosidodecahedron (also known as the small complex ditrigonal rhombicosidodecahedron) is a degenerate uniform star polyhedron. It has 62 faces (20 triangles, 12 pentagrams and 30 squares), 120 (doubled) edges and 20 vertices. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as a topological polyhedron. It can be constructed from the vertex figure 3(/.4.3.4), thus making it also a cantellated great icosahedron. The ""3"" in front of this vertex figure indicates that each vertex in this degenerate polyhedron is in fact three coincident vertices."
"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"
"distinct rectangular faces have edges (""a"", ""b""), (""a"", ""c""), and (""b"", ""c""); so the respective face diagonals have lengths formula_1 formula_2 and formula_3 Thus each face diagonal of a cube with side length ""a"" is formula_4. A regular dodecahedron has 60 face diagonals (and 100 space diagonals). Face diagonal In geometry, a face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to a space diagonal passing through the interior of the polyhedron. A cuboid has twelve face diagonals (two on each of the six faces), and it has four space diagonals. The cuboid's face"
"Trapezo-rhombic dodecahedron In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has ""D"" symmetry. This polyhedron could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts. A space-filling tessellation, the trapezo-rhombic dodecahedral honeycomb, can be made by translated copies of this cell. Each ""layer"" is a hexagonal tiling, or a rhombille tiling, and alternate layers are connected"