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Hexahemioctacron
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4 , 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 triangular portions of each are visible. Related polyhedra It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the square faces in common), and with the octahemioctahedron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 verti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubohemioctahedron
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4 , 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 triangular portions of each are visible. Related polyhedra It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the square faces in common), and with the octahemioctahedron (having the hexagonal faces in common). Tetrahexagonal tiling The ''cubohemioctahedron'' can be seen as a net on the hyperbolic tetrahexagonal tiling with vertex figure 4.6.4.6. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahemioctahedron
In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center. Orientability It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero (a topological torus). Related polyhedra It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common). By Wythoff construction it has tetrahedral symmetry (Td), like the ''rhombitetratetrahedron'' construction for the cuboctahedron, with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry (Oh). In this respect it is akin to the Morin surface, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemipolyhedron
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, and their count is half the number of faces of that other polyhedron – hence the "hemi" prefix. The prefix "hemi" is also used to refer to certain projective polyhedra, such as the hemicube (geometry), hemi-cube, which are the image of a 2 to 1 map of a spherical polyhedron with central symmetry. Wythoff symbol and vertex figure Their Wythoff symbols are of the form ''p''/(''p'' − ''q'') ''p''/''q'' , ''r''; their vertex figures are quadrilateral#More quadrilaterals, crossed quadrilaterals. They are thus related to the cantellation (geometry), cantellated polyhedra, which have similar Wythoff symbols. The vertex configuration is ''p''/''q''.2''r''.''p''/(''p'' − ''q'').2''r''. The 2''r''-gon faces pass through the center of the model: if represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahemioctacron
In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center. Orientability It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero (a topological torus). Related polyhedra It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common). By Wythoff construction it has tetrahedral symmetry (Td), like the ''rhombitetratetrahedron'' construction for the cuboctahedron, with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry (Oh). In this respect it is akin to the Morin surface, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Hemipolyhedra
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, and their count is half the number of faces of that other polyhedron – hence the "hemi" prefix. 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. Wythoff symbol and vertex figure Their Wythoff symbols are of the form ''p''/(''p'' − ''q'') ''p''/''q'' , ''r''; their vertex figures are crossed quadrilaterals. They are thus related to the cantellated polyhedra, which have similar Wythoff symbols. The vertex configuration is ''p''/''q''.2''r''.''p''/(''p'' − ''q'').2''r''. The 2''r''-gon faces pass through the center of the model: if represented as faces of spherical polyhedra, they cover an entire hemisphere and their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahemioctahedron
In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center. Orientability It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero (a topological torus). Related polyhedra It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common). By Wythoff construction it has tetrahedral symmetry (Td), like the ''rhombitetratetrahedron'' construction for the cuboctahedron, with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry (Oh). In this respect it is akin to the Morin surface, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubohemioctahedron
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4 , 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 triangular portions of each are visible. Related polyhedra It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the square faces in common), and with the octahemioctahedron (having the hexagonal faces in common). Tetrahexagonal tiling The ''cubohemioctahedron'' can be seen as a net on the hyperbolic tetrahexagonal tiling with vertex figure 4.6.4.6. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemicube (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] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers. Oblique prism An oblique prism is a pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnus Wenninger
Father Magnus J. Wenninger Order of Saint Benedict, OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to German people, German immigrants in Park Falls, Wisconsin, Joseph Wenninger always knew he was going to be a priest. From an early age, it was understood that his brother Heinie would take after their father and become a baker, and that Joe, as he was then known, would go into the priesthood. When Wenninger was thirteen, after graduating from the parochial school in Park Falls, Wisconsin, his parents saw an advertisement in the German newspaper ''Der Wanderer'' that would help to shape the rest of his life. The ad was for a preparatory school in Collegeville, Minnesota, associated with the Benedictine College of Saint Benedict and Saint John's University, St. John's University. While admitting to feeling homesick a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point (geometry), point where two or more curves, line (geometry), lines, or edge (geometry), edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron, polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a vertex is called "convex set, convex" if the internal an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
