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Nonconvex Polyhedra
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 polyhedron, Kepler–Poinsot polyhedra, 14 Quasiregular polyhedron#Nonconvex examples, quasiregular ones, and 39 semiregular ones. There are also two infinite sets of Uniform_polyhedron#.28p_2_2.29_Prismatic_.5Bp.2C2.5D.2C_I2.28p.29_family_.28Dph_dihedral_symmetry.29, ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have density (polytope), polygon density greater than 1) correspond to circular polygons with overlapping Tessellation, tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex sequence. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book ''Mathematical Models (Cundy and Rollett), Mathematical Models''.Laughlin (2014), p. 16 For uniform polyhedron, uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chirality (mathematics), Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) For example, "" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual polyhedron, dual to an octahedron. The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. Details Chiral and full (or achiral) octahedral symmetry are the Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the Crystal system#Overview of point groups by crystal system, crystallographic point groups of the cubic crystal system. As the hyperoctahedral group of dimension 3 the full octah ... [...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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex. The value means the vertex angle is of the half-circle. "2" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a ''non''-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Symmetry
image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating group, alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere). They are among the Crystal system#Overview of point groups by crystal system, crystallographic point gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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). The tetrahemihexahedron 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.) The tetrahemihexahedron 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prismatic Uniform Polyhedron
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids. Vertex configuration and symmetry groups Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group. The difference between the prismatic and antiprismatic symmetry groups is that D''p''h has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its ''p''-fold axis (parallel to the polygon); while D''p''d has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has ''p'' reflection planes which contain the ''p''-fold axis. The D''p''h symmetry group contains inversion if and only if ''p'' is even, while D''p''d contains inversion symmetry if and only if ''p'' is odd. Enumeration There are: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangles
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "oblong" is used to refer to a non-square rectangle. A rectangle with vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
