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Tetrahemihexahedron
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron). It is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 , 2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.) It is a hemipolyhedron. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence ''hemihexahedron''. Hemi faces are also oriented in the same direction as the regular polyhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboctahedron
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, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was probably known to Plato: Heron's ''Definitiones'' quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Synonyms *''Vector Equilibrium'' (Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry). Fuller also called a cuboctahedron built of rigid struts and flexible vertices a ''jitterbug''; this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its squa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Plane
In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in passing through the origin. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a square by g ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemi-cuboctahedron
A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron. It has 4 triangular faces and 3 square faces, 12 edges, and 6 vertices. It can be seen as a rectified hemi-octahedron or rectified hemi-cube. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles and 3 square), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected. Dual Its dual polyhedron is a rhombic hemi-dodecahedron which has 7 vertices (1-7), 12 edges (a-l), and 6 rhombic faces (A-F). : Related polyhedra It has a real presentation as a uniform star polyhedron, the tetrahemihexahedron. : 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 plane by 10 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cupola (geometry)
In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively. A cupola can be seen as a prism where one of the polygons has been collapsed in half by merging alternate vertices. A cupola can be given an extended Schläfli symbol representing a regular polygon joined by a parallel of its truncation, or Cupolae are a subclass of the prismatoids. Its dual contains a shape that is sort of a weld between half of an -sided trapezohedron and a -sided pyramid. Examples The above-mentioned three polyhedra are the only non-trivial convex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
