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Regular Tilings
This article lists the regular polytopes in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson (mathematician), Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the Edge (geometry), edges of polygons and other higher dimensional polytopes. It is used in the definition of prism (geometry), uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex polygon, convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 Dodecahedral Honeycomb
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) of hyperbolic 3-space. With Schläfli symbol it has four regular dodecahedron, dodecahedra around each Edge (geometry), edge, and 8 dodecahedra around each Vertex (geometry), vertex in an octahedron, octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual polytope, dual is the order-5 cubic honeycomb. Description The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge. Symmetry It has a half symmetry construction, , with two types (colors) of dodecahedra in the Wythoff construction. ↔ . Images A view of the order-4 dodecahed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...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'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 regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertex (geometry), vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two Face (geometry), faces (or polyhedron sides) meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. An ''edge'' may also be an infinite line (geometry), line separating two half-planes. The ''sides'' of a plane angle are semi-infinite Half-line (geometry), half-lines (or rays). Relation to edges in graphs In graph theory, an Edge (graph theory), edge is an abstract object connecting two vertex (graph theory), graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its n-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Johnson (mathematician)
Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was an American mathematician at Wheaton College, Norton, Massachusetts. Early life and education Norman Johnson was born on in Chicago. His father had a bookstore and published a local newspaper. Johnson earned his undergraduate mathematics degree in 1953 at Carleton College in Northfield, Minnesota followed by a master's degree from the University of Pittsburgh. After graduating in 1953, Johnson did alternative civilian service as a conscientious objector. He earned his PhD from the University of Toronto in 1966 with a dissertation title of ''The Theory of Uniform Polytopes and Honeycombs'' under the supervision of H. S. M. Coxeter. From there, he accepted a position in the Mathematics Department of Wheaton College in Massachusetts and taught until his retirement in 1998. Career In 1966, he enumerated 92 convex non-uniform polyhedra with regular faces. Victor Zalgaller later proved (1969) that Johnson's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean space, 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 definition, recursive description, starting with \ for a p-sided regular polygon that is Convex set, convex. For example, is an equilateral triangle, is a Square (geometry), 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 pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including '' The Fifty-Nine Icosahedra'' (1938) and '' Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane (mathematics), plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudosphere, pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they local property, locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of event (relativity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
