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Icosahedron
In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/[1]) is a polyhedron with 20 faces. The name comes from Greek εἴκοσι (eíkosi), meaning 'twenty', and ἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons". There are many kinds of icosahedra, with some being more symmetrical than others
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Prism
In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. At least two of the flat surfaces must have an angle between them. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type. Some types of optical prism are not in fact in the shape of geometric prisms. Prisms can be made from any material that is transparent to the wavelengths for which they are designed. Typical materials include glass, plastic, and fluorite. A dispersive prism can be used to break light up into its constituent spectral colors (the colors of the rainbow)
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Right Angle
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees),[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[4] making the right angle basic to trigonometry.Contents1 In elementary geometry 2 Symbols 3 Euclid 4 Conversion to other units 5 Rule of 3-4-5 6 Thales' theorem 7 See also 8 ReferencesIn elementary geometry[edit] A rectangle is a quadrilateral with four right angles
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Vertex (geometry)
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet
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Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.[1] In a polygon, an edge is a line segment on the boundary,[2] and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet.[3] A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.Contents1 Relation to edges in graphs 2 Number of edges in a polyhedron 3 Incidences with other faces 4 Alternative terminology 5 See also 6 References 7 External linksRelation to edges in graphs[edit] In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment
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Face (geometry)
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object;[1] a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).[2]Contents1 Polygonal face1.1 Number of polygonal faces of a polyhedron2 k-face2.1 Cell or 3-face 2.2 Facet or (n-1)-face 2.3 Ridge or (n-2)-face 2.4 Peak or (n-3)-face3 See also 4 References 5 External linksPolygonal face[edit] In elementary geometry, a face is a polygon on the boundary of a polyhedron.[2][3] Other names for a polygonal face include side of a polyhedron, and tile of a Euclidean plane tessellation. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope
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Coxeter Diagram
In geometry, a Coxeter– Dynkin diagram
Dynkin diagram
(or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge). An unlabeled branch implicitly represents order-3. Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6
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Geometry
Geometry
Geometry
(from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes
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Convex Set
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.[1] More specifically, in a Euclidean space, a convex region is a region where, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region.[2][3] For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space
Euclidean space
is called the convex hull of A. It is the smallest convex set containing A. A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set
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Net (polyhedron)
In geometry a net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.[1] An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book Unterweysung der Messung mit dem Zyrkel und Rychtscheyd included nets for the Platonic solids and several of the Archimedean solids.[2]Contents1 Existence and uniqueness 2 Shortest path 3 Higher-dimensional polytope nets 4 See also 5 References 6 External linksExistence and uniqueness[edit] Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated
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Alternation (geometry)
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.[1] Coxeter
Coxeter
labels an alternation by a prefixed by an h, standing for hemi or half. Because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons
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Truncated Octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces
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Dual Uniform Polyhedron
A dual uniform polyhedron is the dual of a uniform polyhedron. Just like a uniform polyhedron is vertex-transitive, a dual uniform polyhedron is face-transitive.Uniform set Dual uniform set5 Platonic solids 5 Platonic solids4 Kepler solids 4 Kepler solids13 Archimedean solids 13 Catalan solid∞ Prisms ∞ Bipyramids∞ Antiprisms ∞ Trapezohedra53 nonregular uniform star polyhedra 53 dual uniform star polyhedraExample[edit]List of uniform polyhedraReferences[edit]Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.  Wenninger, Magnus (1983). Dual Models
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Dihedral Angle
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge
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Hilbert's Third Problem
The third on Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Gauss,[1] Hilbert conjectured that this is not always possible
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