1.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2

2.
Uniform star polyhedron
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In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting, each polyhedron can contain either star polygon faces, star polygon vertex figures or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra,5 quasiregular ones, there are also two infinite sets of uniform star prisms and uniform star antiprisms. The nonconvex forms are constructed from Schwarz triangles, all the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements. Regular polyhedra are labeled by their Schläfli symbol, other nonregular uniform polyhedra are listed with their vertex configuration or their Uniform polyhedron index U. Note, For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull vertex arrangement has same topology as one of these, for example an nonuniform cantellated form may have rectangles created in place of the edges rather than squares. There is one form, the tetrahemihexahedron which has tetrahedral symmetry. There are two Schwarz triangles that generate unique nonconvex uniform polyhedra, one triangle, and one general triangle. The general triangle generates the octahemioctahedron which is given further on with its octahedral symmetry. There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry, there are four Schwarz triangles that generate nonconvex forms, two right triangles, and, and two general triangles. There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry, some of the nonconvex snub forms have reflective vertex symmetry. Coxeter identified a number of star polyhedra by the Wythoff construction method. It is counted as a uniform polyhedron rather than a uniform polyhedron because of its double edges. Star polygon List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, a proof of the completeness on the list of elementary homogeneous polyhedra, Ukrainskiui Geometricheskiui Sbornik, 139–156, MR0326550 Skilling, J. The complete set of polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences,278, 111–135, doi,10. 1098/rsta.1975.0022, ISSN 0080-4614, JSTOR74475, MR0365333 HarEl, zvi Har’El, Kaleido software, Images, dual images Mäder, R. E. Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals

3.
Cubohemioctahedron
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In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral and it is given Wythoff symbol 4/34 |3, although that is a double-covering of this figure. A nonconvex polyhedron has intersecting faces which do not represent new edges or faces, in the picture vertices are marked by golden spheres, and edges by silver cylinders. It is a hemipolyhedron with 4 hexagonal faces passing through the model center, the hexagons intersect each other and so only triangle portions of each are visible. It shares the vertex arrangement and edge arrangement with the cuboctahedron, the cubohemioctahedron can be seen as a net on the hyperbolic tetrahexagonal tiling with vertex figure 4.6.4.6. The hexahemioctacron is the dual of the cubohemioctahedron, and is one of nine dual hemipolyhedra and it appears visually indistinct from the octahemioctacron. Since the cubohemioctahedron has four hexagonal faces passing through the center, thus it is degenerate. In Magnus Wenningers Dual Models, they are represented with intersecting infinite prisms passing through the model center, hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron. Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Weisstein, Eric W. Weisstein, Cubohemioctahedron at MathWorld

4.
Euler characteristic
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It is commonly denoted by χ. The Euler characteristic was originally defined for polyhedra and used to prove theorems about them. Leonhard Euler, for whom the concept is named, was responsible for much of early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, any convex polyhedrons surface has Euler characteristic V − E + F =2. This equation is known as Eulers polyhedron formula and it corresponds to the Euler characteristic of the sphere, and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below and this version holds both for convex polyhedra and the non-convex Kepler-Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0. The Euler characteristic can be defined for connected plane graphs by the same V − E + F formula as for polyhedral surfaces, the Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of determined by G. For trees, E = V −1 and F =1, if G has C components, the same argument by induction on F shows that V − E + F − C =1. One of the few graph theory papers of Cauchy also proves this result, via stereographic projection the plane maps to the two-dimensional sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchys proof of Eulers formula given below, there are many proofs of Eulers formula. One was given by Cauchy in 1811, as follows and it applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface, after this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, therefore, proving Eulers formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that arent connected yet. This adds one edge and one face and does not change the number of vertices, continue adding edges in this manner until all of the faces are triangular. This decreases the number of edges and faces by one each and does not change the number of vertices, remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph

5.
Hemipolyhedron
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In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These hemi faces lie parallel to the faces of some other symmetrical polyhedron, the prefix hemi is also used to refer to certain projective polyhedra, such as the hemi-cube, which are the image of a 2 to 1 map of a spherical polyhedron with central symmetry. Their Wythoff symbols are of the form p/ p/q | r and they are thus related to the cantellated polyhedra, which have similar Wythoff symbols. The 2r-gon faces pass through the center of the model, if represented as faces of spherical polyhedra, they cover an entire hemisphere, the p/ notation implies a face turning backwards around the vertex figure. The nine forms, listed with their Wythoff symbols and vertex configurations are, only the octahemioctahedron represents an orientable surface, the remaining hemipolyhedra have non-orientable or single-sided surfaces. Since the hemipolyhedra have faces passing through the center, the figures have corresponding vertices at infinity, properly. In Magnus Wenningers Dual Models, they are represented with intersecting prisms, in practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, however, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions. There are 9 such duals, sharing only 5 distinct outward forms, the members of a given identical pair differ in their arrangements of true and false vertices. The outwards forms are, The hemipolyhedra occur in pairs as facetings of the polyhedra with four faces at a vertex. These quasiregular polyhedra have vertex configuration m. n. m. n and their edges, in addition to forming the m- and n-gonal faces, thus, the hemipolyhedra can be derived from the quasiregular polyhedra by discarding either the m-gons or n-gons and then inserting the hemi faces. Since the hemipolyhedra, like the quasiregular polyhedra, also have two types of faces alternating around each vertex, they are also considered to be quasiregular. Here m and n correspond to p/q above, and h corresponds to 2r above, there are also related Euclidean tilings based on quasiregular tilings, that use apeirogons as the equivalents of the diametral 2r-gons above. Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, uniform polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences, The Royal Society,246, 401–450, doi,10. 1098/rsta.1954

6.
Octahemioctahedron
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In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as U3. Its vertex figure is a crossed quadrilateral and it is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center. It is the only hemipolyhedron that is orientable, and the uniform polyhedron with an Euler characteristic of zero. It shares the vertex arrangement and edge arrangement with the cuboctahedron, by Wythoff construction it has tetrahedral symmetry, like the rhombitetratetrahedron construction for the cuboctahedron, with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry, the octahemioctacron is the dual of the octahemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the hexahemioctacron, since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity, properly, on the real projective plane at infinity. In Magnus Wenningers Dual Models, they are represented with intersecting prisms, in practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, however, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions. The octahemioctacron has four vertices at infinity, compound of five octahemioctahedra Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron. Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Eric W. Weisstein, Octahemioctahedron at MathWorld