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Hosohedron
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle radians ( degrees). Hosohedra as regular polyhedra For a regular polyhedron whose Schläfli symbol is , the number of polygonal faces is : :N_2=\frac. The Platonic solids known to antiquity are the only integer solutions for ''m'' ≥ 3 and ''n'' ≥ 3. The restriction ''m'' ≥ 3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing ''m'' = 2 makes :N_2=\frac=n, and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron is represented as ''n'' abutting lunes, with interior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonal Hosohedron
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle radians ( degrees). Hosohedra as regular polyhedra For a regular polyhedron whose Schläfli symbol is , the number of polygonal faces is : :N_2=\frac. The Platonic solids known to antiquity are the only integer solutions for ''m'' ≥ 3 and ''n'' ≥ 3. The restriction ''m'' ≥ 3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing ''m'' = 2 makes :N_2=\frac=n, and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron is represented as ''n'' abutting lunes, with interi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form , where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, and five regular compounds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Lune
In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, θ, with dihedral angle θ. The word "lune" derives from ''luna'', the Latin word for Moon. Properties Great circles are the largest possible circles (circumferences) of a sphere; each one divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points. Common examples of great circles are lines of longitude (''meridians'') on a sphere, which meet at the north and south poles. A spherical lune has two planes of symmetry. It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles. Surface area The surface area of a spherical lune is 2θ ''R''2, where ''R'' is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles. When this angle eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold Notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic plane (H^2). Definition of the notation The following types of Euclidean transformation can occur in a group described by orbifold notation: * reflection through a line (or plane) * translation by a vector * rotation of finite order around a point * infinite rotation around a line in 3- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol . It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated ''digon'', t is a square, . An alternated digon, h is a monogon, . In Euclidean geometry The digon can have one of two visual representations if placed in Euclidean space. One representation is degenerate, and visually appears as a double-covering of a line segment. Appearing when the minimum distance between the two edges is 0, this form arises in several situations. This double-covering form is sometimes used for defining degener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the ''Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Tiling
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron. Some "improper" polyhedra, such as hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron. History The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, and appear to date from the neolithic period (the New Stone Age). During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) wrote the first ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedron
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of a lens space L(''p'',''q''). Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons. As a spherical tiling, a dihedron can exist as nondegenerate form, with two ''n''-sided faces covering the sphere, each face being a hemisphere, and vertices on a great circle. It is regular if the vertices are equally spaced. The dual of an ''n''-gonal dihedron is an ''n''-gonal hosohedron, where ''n'' digon faces share two vertices. As a flat-faced polyhedron A dihedron can be considered a degenerate prism whose two (planar) ''n''-sided polygon bases are connected "back-to-back", so that the resulting object has no depth. The polygons must b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinmetz Solid
In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is called a bicylinder. Topologically, it is equivalent to a square hosohedron. The intersection of three cylinders is called a tricylinder. A bisected bicylinder is called a vault, and a cloister vault in architecture has this shape. Steinmetz solids are named after mathematician Charles Proteus Steinmetz, who solved the problem of determining the volume of the intersection. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. Bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac r^3 and the ;surface area :A=16 r^2. The upper half of a bicylinder is the square ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. Definition The Schläfli symbol is a recursive description, starting with for a ''p''-sided regular polygon that is convex. For example, is an equilateral triangle, is a square, a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols contain irreducible fractions ''p''/''q'', where ''p'' is the number of vertices, and ''q'' is their turning number. Equivalently, is created from the vertices of , connected every ''q''. For example, is a pentagram; is a pentagon. A regular polyhedron that has ''q'' regular ''p''-sided Face (geometry), polygon faces around each Verte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not refer to a face but to the internal polygon base, lying in the mirror plane that connects the two pyramid halves. (If it were a face, then each of its edges would connect three faces instead of two.) "Regular", right bipyramids A ''"regular"'' bipyramid has a ''regular'' polygon base. It is usually implied to be also a ''right'' bipyramid. A ''right'' bipyramid has its two apices ''right'' above and ''right'' below the center or the ''centroid'' of its polygon base. A "regular" right (symmetric) -gonal bipyramid has Schläfli symbol . A right (symmetric) bipyramid has Schläfli symbol , for polygon base . The "regular" right (thus face-transitive) -gonal bipyramid with regular vertices is the dual of the -gonal uniform (thus right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
