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Self-intersecting Polygon
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edge (geometry), edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: *the crossed quadrilateral, with four edges **the antiparallelogram, a crossed quadrilateral with alternate edges of equal length ***the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel *Star polygons **pentagram, with five edges **heptagram, with seven edges **octagram, with eight edges **enneagram (geometry), enneagram or nonagram, with nine edges **decagram (geometry), decagram, with ten edges **hendecagram, with eleven edges **dodecagram, with twelve edges See also *List of regular polytopes and compounds#Stars *Complex polygon {{DEFAULTSORT:Self-intersecting polygons, Geometric shapes Mathematics-related lists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. 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. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagram
In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line". Detail In general, an octagram is any self-intersecting octagon (8-sided polygon). The regular octagram is labeled by the Schläfli symbol , which means an 8-sided star, connected by every third point. Variations These variations have a lower dihedral, Dih4, symmetry: The symbol Rub el Hizb is a Unicode glyph ۞ at U+06DE. As a quasitruncated square Deeper truncations of the square can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an octagon, t=. A quasitruncated square, inverted as , is an octagram, t=.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Polygon
The term ''complex polygon'' can mean two different things: * In geometry, a polygon in the unitary plane, which has two complex dimensions. * In computer graphics, a polygon whose boundary is not simple. Geometry In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions. A complex number may be represented in the form (a + ib), where a and b are real numbers, and i is the square root of -1. Multiples of i such as ib are called ''imaginary numbers''. A complex number lies in a complex plane having one real and one imaginary dimension, which may be represented as an Argand diagram. So a single complex dimension comprises two spatial dimensions, but of different kinds - one real and the other imaginary. The unitary plane comprises two such complex planes, which are orthogonal to each other. Thus it has two real dimensions and two imaginary dimensions. A complex polygon is a (complex) two-dimensional (i.e. four spatial dimensions) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes And Compounds
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, ,3or , it is represented by Coxeter diagram . The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecagram
In geometry, a dodecagram ( Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus) is a or compound with 12 vertices. There is one regular dodecagram polygon (with and a turning number of 5). There are also 4 regular compounds and Regular dodecagram There is one regula ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendecagram
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices. The name ''hendecagram'' combines a Greek numeral prefix, '' hendeca-'', with the Greek suffix ''-gram''. The ''hendeca-'' prefix derives from Greek ἕνδεκα (ἕν + δέκα, one + ten) meaning "eleven". The ''-gram'' suffix derives from γραμμῆς (''grammēs'') meaning a line. Regular hendecagrams There are four regular hendecagrams, which can be described by the notation , , , and ; in this notation, the number after the slash indicates the number of steps between pairs of points that are connected by edges. These same four forms can also be considered as stellations of a regular hendecagon. Since 11 is prime, all hendecagrams are star polygons and not compound figures. Construction As with all odd regular polygons and star polygons whose orders are not products of distinct Fermat primes, the regular hendecagrams cannot be constructed with compass and stra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagram (geometry)
In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is . The name ''decagram'' combines a numeral prefix, ''deca-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from ''γραμμῆς'' (''grammēs'') meaning a line. Regular decagram For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below. Applications Decagrams have been used as one of the decorative motifs in girih tiles. : Isotoxal variations An isotoxal polygon has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom of one angle. The first is a variation of a double-wound of a pentagon , and last is a variation of a double-wound of a pentagram . The middle is a variation of a regular decagram, . Related figures A regular decagram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enneagram (geometry)
In geometry, an enneagram (🟙 U+1F7D9) is a nine-pointed plane figure. It is sometimes called a nonagram, nonangle, or enneagon. The word 'enneagram' combines the numeral prefix ''ennea-'' with the Greek suffix '' -gram''. The ''gram'' suffix derives from ''γραμμῆς'' (''grammēs'') meaning a line. Regular enneagram A regular enneagram is a 9-sided star polygon. It is constructed using the same points as the regular enneagon, but the points are connected in fixed steps. Two forms of regular enneagram exist: *One form connects every second point and is represented by the Schläfli symbol . *The other form connects every fourth point and is represented by the Schläfli symbol . There is also a star figure, or 3, made from the regular enneagon points but connected as a compound of three equilateral triangles. (If the triangles are alternately interlaced, this results in a Brunnian link.) This star figure is sometimes known as the ''star of Goliath'', after or 2, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagram
A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes. The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from ''γραμμῆ'' (''grammē'') meaning a line. Geometry In general, a heptagram is any self-intersecting heptagon (7-sided polygon). There are two ''regular'' heptagrams, labeled as and , with the second number representing the vertex interval step from a regular heptagon, . This is the smallest star polygon that can be drawn in two forms, as irreducible fractions. The two heptagrams are sometimes called the ''heptagram'' (for ) and the ''great heptagram'' (for ). The previous one, the regular hexagram , is a compound of two triangles. The smallest star polygon is the pentagram. The next one is the octagram and its related star figure (a compound of two squares), followed by the regular enneagram, which also has two forms: and , as wel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (geometry)
In geometry, an edge is a particular type of line segment joining two 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 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. Relation to edges in graphs In graph theory, an edge is an abstract object connecting two 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 skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle. Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the Five Holy Wounds, five wounds of Jesus. Today the symbol is widely used by the Wiccans, witches, and pagans. The pentagram has Magic (supernatural), magical associations. Many people who practice neopaganism wear jewelry incorporating the symbol. The word ''pentagram'' comes from the Greek language, Greek word πεντάγραμμον (''pentagrammon''), fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple and star polygons. Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons. The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having 2 or more turns (turning number and density), like in spirolaterals.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p.24 Etymology Star polygon names combine a numeral prefix, such as ''penta-'', with the Greek suffix '' -gram'' (in this cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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