1.
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

2.
Regular 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

3.
Regular icosahedron
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In geometry, a regular icosahedron is a convex polyhedron with 20 faces,30 edges and 12 vertices. It is one of the five Platonic solids, and also the one with the most sides and it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35 and it is the dual of the dodecahedron, which is represented by, having three pentagonal faces around each vertex. A regular icosahedron is a pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedrons or icosahedra. The surface area A and the volume V of a regular icosahedron of edge length a are, note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ −1, the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, if two vertices are taken to be at the north and south poles, then the other ten vertices are at latitude ±arctan ≈ ±26. 57°. These ten vertices are at evenly spaced longitudes, alternating between north and south latitudes and 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 icosahedron has 43,380 distinct nets. To color the icosahedron, such that no two adjacent faces have the color, requires at least 3 colors. A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere, the problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers, as it turns out, the icosahedron occupies less of the spheres volume than the dodecahedron. The following construction of the icosahedron avoids tedious computations in the number field ℚ necessary in more elementary approaches, the existence of the icosahedron amounts to the existence of six equiangular lines in ℝ3. Indeed, intersecting such a system of lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of an icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. In order to such an equiangular system, we start with this 6 ×6 square matrix

4.
Dual graph
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In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, the definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a generalization of the geometric concepts of dual polyhedra and dual tessellations. Variations of planar graph duality include a version of duality for directed graphs, however, these notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term dual is used because the property of being a graph is symmetric, meaning that if H is a dual of a connected graph G. When discussing the dual of a graph G, the graph G itself may be referred to as the primal graph, many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have also applied in computer vision, computational geometry, mesh generation. The unique planar embedding of a cycle graph divides the plane into two regions, the inside and outside of the cycle, by the Jordan curve theorem. However, in an n-cycle, these two regions are separated from each other by n different edges, therefore, the dual graph of the n-cycle is a multigraph with two vertices, connected to each other by n dual edges. Such a graph is called a dipole graph, conversely, the dual to an n-edge dipole graph is an n-cycle. According to Steinitzs theorem, every polyhedral graph must be planar and 3-vertex-connected, whenever two polyhedra are dual, their graphs are also dual. For instance the Platonic solids come in pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron. Polyhedron duality can also be extended to duality of higher dimensional polytopes, a plane graph is said to be self-dual if it is isomorphic to its dual graph. The wheel graphs provide a family of self-dual graphs coming from self-dual polyhedra. However, there also exist self-dual graphs that are not polyhedral and it follows from Eulers formula that every self-dual graph with n vertices has exactly 2n −2 edges

5.
Antiprism graph
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In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges and they are regular, polyhedral, and also Hamiltonian graphs. The first graph in the sequence, the graph, has 6 vertices and 12 edges. Although geometrically the star polygons also form the faces of a different sequence of antiprisms, an antiprism graph is a special case of a circulant graph, Ci2n. Other infinite sequences of polyhedral graph formed in a way from polyhedra with regular-polygon bases include the prism graphs. Other vertex-transitive polyhedral graphs include the Archimedean graphs

6.
Prism graph
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In the mathematical field of graph theory, a prism graph is a graph that has one of the prisms as its skeleton. Prism graphs are examples of generalized Petersen graphs, with parameters GP and they may also be constructed as the Cartesian product of a cycle graph with a single edge. As with many vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs, the order-n dihedral group is the group of symmetries of a regular n-gon in the plane, it acts on the n-gon by rotations and reflections. It can be generated by two elements, a rotation by an angle of 2π/n and a rotation, and its Cayley graph with this generating set is the prism graph. Abstractly, the group has the presentation ⟨ r, f ∣ r n, f 2,2 ⟩, the n-gonal prism graphs with odd values of n may be constructed as circulant graphs C2 n 2, n. However, this construction does not work for even values of n, the graph of an n-gonal prism has 2n vertices and 3n edges. Since the prism has symmetries taking each vertex to each other vertex, as polyhedral graphs, they are also 3-vertex-connected planar graphs. Every prism graph has a Hamiltonian cycle, among all biconnected cubic graphs, the prism graphs have within a constant factor of the largest possible number of 1-factorizations. A 1-factorization is a partition of the set of the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every biconnected n-vertex cubic graph has O 1-factorizations, and the graphs have Ω 1-factorizations. The number of spanning trees of a prism graph is given by the formula n 2 For n =3,4,5. These numbers are 75,384,1805,8100,35287,150528, the n-gonal prism graphs for even values of n are partial cubes. They form one of the few known infinite families of partial cubes. The pentagonal prism is one of the minors for the graphs of treewidth three. The triangular prism and cube graph have treewidth exactly three, but all larger prism graphs have treewidth four, other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the antiprism graphs and wheel graphs. Other vertex-transitive polyhedral graphs include the Archimedean graphs, if the two cycles of a prism graph are broken by the removal of a single edge in the same position in both cycles, the result is a ladder graph. If these two removed edges are replaced by two crossed edges, the result is a graph called a Möbius ladder

7.
Four color theorem
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Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers, according to an article by the math historian Kenneth May, “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. Martin Gardner wrote an account of what was known at the time about the four color theorem in his September 1960 Mathematical Games column in Scientific American magazine. In 1975 Gardner revisited the topic by publishing a map said to be a counter-example in his infamous April fools hoax column of April 1975, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer, Appel and Hakens approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exist because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. To dispel remaining doubt about the Appel–Haken proof, a proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour. Additionally, in 2005, the theorem was proved by Georges Gonthier with general purpose theorem proving software, the intuitive statement of the four color theorem, i. e. First, all corners, points that belong to three or more countries, must be ignored. In addition, bizarre maps can require more than four colors, second, for the purpose of the theorem, every country has to be a connected region, or contiguous. In the real world, this is not true, because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map, In this map and this map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. A similar construction also applies if a color is used for all bodies of water. For maps in which more than one country may have multiple disconnected regions, a simpler statement of the theorem uses graph theory

8.
Left-right planarity test
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In a 2003 experimental comparison of six planarity testing algorithms, this was one of the fastest algorithms tested. For any depth-first search of a graph G, the edges encountered when discovering a vertex for the first time define a depth-first search tree T of G. This is a Trémaux tree, meaning that the edges each connect a pair of vertices that are related to each other as an ancestor. Three types of patterns can be used to define two relations between pairs of edges, named the T-alike and T-opposite relations. In the following figures, simple circle nodes represent vertices, double circle nodes represent subtrees, twisted segments represent tree paths, the root of each tree is shown at the bottom of the figure. In the first figure, the edges labeled α and β are T-alike, meaning that, at the endpoints nearest the root of the tree, they will both be on the same side of the tree in every planar drawing. In the next two figures, the edges with the labels are T-opposite, meaning that they will be on different sides of the tree in every planar drawing