1.
Convex hull
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In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space is the smallest convex set that contains X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces, they may also be generalized further, to oriented matroids. The algorithmic problem of finding the convex hull of a set of points in the plane or other low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry. A set of points is defined to be if it contains the line segments connecting each pair of its points. The convex hull of a given set X may be defined as The minimal convex set containing X The intersection of all convex sets containing X The set of all combinations of points in X. The union of all simplices with vertices in X and it is not obvious that the first definition makes sense, why should there exist a unique minimal convex set containing X, for every X. Thus, it is exactly the unique minimal convex set containing X. Each convex set containing X must contain all convex combinations of points in X, in fact, according to Carathéodorys theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N +1 points are sufficient in the definition above. If the convex hull of X is a set, then it is the intersection of all closed half-spaces containing X. The hyperplane separation theorem proves that in case, each point not in the convex hull can be separated from the convex hull by a half-space. However, there exist convex sets, and convex hulls of sets, more abstractly, the convex-hull operator Conv has the characteristic properties of a closure operator, It is extensive, meaning that the convex hull of every set X is a superset of X. It is non-decreasing, meaning that, for two sets X and Y with X ⊆ Y, the convex hull of X is a subset of the convex hull of Y. It is idempotent, meaning that for every X, the hull of the convex hull of X is the same as the convex hull of X. The convex hull of a point set S is the set of all convex combinations of its points. For each choice of coefficients, the convex combination is a point in the convex hull. Expressing this as a formula, the convex hull is the set. The convex hull of a point set S ⊊ R n forms a convex polygon when n =2. Each point x i in S that is not in the hull of the other points is called a vertex of Conv . In fact, every convex polytope in R n is the hull of its vertices

2.
Truncated icosidodecahedron
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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 30 square faces,20 regular hexagonal faces,12 regular decagonal faces,120 vertices and 180 edges – more than any other convex nonprismatic uniform polyhedron, since each of its faces has point symmetry, the truncated icosidodecahedron is a zonohedron. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure, however, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular. One unfortunate point of confusion is there is a nonconvex uniform polyhedron of the same name. The surface area A and the volume V of the truncated icosidodecahedron of edge length a are, V = a 3 ≈206.803399 a 3. If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest. Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ −2, centered at the origin, are all the permutations of, and. The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosidodecahedron can also be represented as a spherical tiling 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, schlegel diagrams are similar, with a perspective projection and straight edges. Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces, the truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases. In the mathematical field of theory, a truncated icosidodecahedral graph is the graph of vertices and edges of the truncated icosidodecahedron. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph and this polyhedron can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p <6, the members of the sequence are omnitruncated polyhedra, for p >6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. Wenninger, Magnus, Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR0467493 Cromwell, the Geometrical Foundation of Natural Structure, A Source Book of Design. Cromwell, P. Polyhedra, CUP hbk, pbk, eric W. Weisstein, GreatRhombicosidodecahedron at MathWorld. 3D convex uniform polyhedra x3x5x - grid, editable printable net of a truncated icosidodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra

3.
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