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Hemi-dodecahedron
A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has 6 pentagonal faces, 15 edges, and 10 vertices. Projections It can be projected symmetrically inside of a 10-sided or 12-sided perimeter: : Petersen graph From the point of view of graph theory this is an embedding of the Petersen graph on a real projective plane. With this embedding, the dual graph is ''K''6 (the complete graph with 6 vertices) --- see hemi-icosahedron. See also * 57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra. *hemi-icosahedron * hemi-cube *hemi-octahedron A hemi-octahedron is an abstract regular polyhedron, containing half ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemi-octahedron
A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron. It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the hemicube. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into four equal parts. It can be seen as a square pyramid without its base. It can be represented symmetrically as a hexagonal or square Schlegel diagram: : It has an unexpected property that there are two distinct edges between every pair of vertices – any two vertices define a digon. See also *Hemi-dodecahedron *Hemi-icosahedron A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemi-dodecahedron2
A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has 6 pentagonal faces, 15 edges, and 10 vertices. Projections It can be projected symmetrically inside of a 10-sided or 12-sided perimeter: : Petersen graph From the point of view of graph theory this is an embedding of the Petersen graph on a real projective plane. With this embedding, the dual graph is ''K''6 (the complete graph with 6 vertices) --- see hemi-icosahedron. See also *57-cell – an abstract regular 4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, a synonym for "spherical polyhedron". However, the term elliptic geometry applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra. As cellular decompositions of the projective plane, they have Euler characteristic 1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with ''locally'' projective polyhedra, which are defined in the theory of abstract polyhedra. Non-overlapping projective polyhedra (density 1) correspond to spherical polyhedra (equivalently, convex poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
57-cell
In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of each cell (60). The symmetry abstract structure is the projective special linear group, L2(19). It has Schläfli symbol with 5 hemi-dodecahedral cells around each edge. It was discovered by . Perkel graph The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array , discovered by . See also * 11-cell – abstract regular polytope with hemi-icosahedral cells. * 120-cell – regular 4-polytope with dodecahedral cells * Order-5 dodecahedral honeycomb - regular hyperbolic honeycomb with same Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a ''realization'' of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory. Introductory concepts Traditional versus abstract polytopes In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example a square and a trapezoid both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. They are said to be isomorphic or “structure preserving”. This common structure may be represented in an underlying abstract polytope, a purely algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Regular Polyhedron
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a ''realization'' of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory. Introductory concepts Traditional versus abstract polytopes In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example a square and a trapezoid both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. They are said to be isomorphic or “structure preserving”. This common structure may be represented in an underlying abstract polytope, a purely algebrai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemi-cube (geometry)
In abstract geometry, a hemicube is an abstract, regular polyhedron, containing half the faces of a cube. Realization It can be realized as a projective polyhedron (a tessellation of the real projective plane by three quadrilaterals), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets. From the point of view of graph theory the skeleton is a tetrahedral graph, an embedding of ''K''4 (the complete graph with four vertices) on a projective plane. The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron. Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be ''cut and unfolded'' as nets in 3-space. Definition A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Double Cover
Petersen is a common Danish patronymic surname, meaning ''"son of Peter"''. There are other spellings. Petersen may refer to: People In arts and entertainment * Adolf Dahm-Petersen, Norwegian voice specialist * Anja Petersen, German operatic soprano and university lecturer * Anker Eli Petersen, Faroese writer and artist * Ann Petersen, Belgian actress * Chris Petersen (born 1963), American child actor * Devon Petersen (born 1986), South African darts player * Elmer Petersen, American artist * Gustaf Munch-Petersen, Danish writer and painter * Joel Petersen, bass guitarist * John Hahn-Petersen, Danish actor * Josef Petersen, Danish novelist * Patrick Petersen, American actor * Paul Petersen, American movie actor, singer, novelist, and activist * Robert E. Petersen, publisher, auto museum founder * Robert Storm Petersen, Danish cartoonist, writer, animator, illustrator, painter and humorist * Sandy Petersen, American game designer * Uwe Fahrenkrog-Petersen, German mus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edges (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge of has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of . The definition of the dual depends on the choice of embedding of the graph , so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. 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 topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by . Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearance in tropical geometry. The cone over the Petersen graph is naturally identif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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