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Octagon
In geometry , an OCTAGON (from the Greek ὀκτάγωνον _oktágōnon_, "eight angles") is an eight-sided polygon or 8-gon. A _regular octagon_ has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square , t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon , t{16}
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Regular Polygon
Regular polygons Edges and vertices _n_ Schläfli symbol {_n_} Coxeter–Dynkin diagram Symmetry group Dn , order 2n Dual polygon Self-dual Area
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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 SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn 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
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place
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Schläfli Symbol
In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol
Schläfli symbol
is named after the 19th-century Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas
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Coxeter Diagram
In geometry , a COXETER–DYNKIN DIAGRAM (or COXETER DIAGRAM, COXETER GRAPH) is a graph with numerically labeled edges (called BRANCHES) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ). An unlabeled branch implicitly represents order-3. Each diagram represents a Coxeter group , and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed , while Coxeter diagrams are undirected ; secondly, Dynkin diagrams must satisfy an additional (crystallographic ) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6
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Dihedral Symmetry
In mathematics , a DIHEDRAL GROUP is the group of symmetries of a regular polygon , which includes rotations and reflections . Dihedral groups are among the simplest examples of finite groups , and they play an important role in group theory , geometry , and chemistry . The notation for the dihedral group of order _n_ differs in geometry and abstract algebra . In geometry , D_n_ or Dih_n_ refers to the symmetries of the n-gon , a group of order 2_n_. In abstract algebra , D_n_ refers to the dihedral group of order _n_. The geometric convention is used in this article
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Internal Angle
2D ANGLES * Right * Interior * Exterior-------------------------2D ANGLE PAIRS * Adjacent * Vertical * Complementary * Supplementary * Transversal -------------------------3D ANGLES * Dihedral Internal and External angles In geometry , an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex , this angle is called an INTERIOR ANGLE (or INTERNAL ANGLE) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex . If every internal angle of a simple polygon is less than 180°, the polygon is called convex . In contrast, an EXTERIOR ANGLE (or EXTERNAL ANGLE) is an angle formed by one side of a simple polygon and a line extended from an adjacent side . :pp
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Degree (angle)
A DEGREE (in full, a DEGREE OF ARC, ARC DEGREE, or ARCDEGREE), usually denoted by ° (the degree symbol ), is a measurement of a plane angle , defined so that a full rotation is 360 degrees. It is not an SI unit , as the SI unit of angular measure is the radian , but it is mentioned in the SI brochure as an accepted unit . Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians. CONTENTS * 1 History * 2 Subdivisions * 3 Alternative units * 4 See also * 5 Notes * 6 References * 7 External links HISTORY See also: Decans
Decans
A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle. The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year
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Dual Polygon
In geometry , polygons are associated into pairs called DUALS, where the vertices of one correspond to the edges of the other. CONTENTS * 1 Properties * 2 Duality in quadrilaterals * 3 Kinds of duality * 3.1 Rectification * 3.2 Polar reciprocation * 3.3 Projective duality * 3.4 Combinatorially * 4 See also * 5 References * 6 External links PROPERTIES Dorman Luke construction , showing a rhombus face being dual to a rectangle vertex figure . Regular polygons are self-dual . The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon , longer sides correspond to larger exterior angles in the dual (a tangential polygon ), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely
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Convex Polygon
A CONVEX POLYGON is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set . In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees
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Cyclic Polygon
In geometry , the CIRCUMSCRIBED CIRCLE or CIRCUMCIRCLE of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the CIRCUMCENTER and its radius is called the CIRCUMRADIUS. A polygon which has a circumscribed circle is called a CYCLIC POLYGON (sometimes a CONCYCLIC POLYGON, because the vertices are concyclic ). All regular simple polygons , all isosceles trapezoids , all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle , which is the smallest circle that completely contains the polygon within it. Not every polygon has a circumscribed circle, as the vertices of a polygon do not need to all lie on a circle, but every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm
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Equilateral Polygon
In geometry , three or more than three straight lines (or segment of a line) make a polygon and an EQUILATERAL POLYGON is a polygon which has all sides of the same length. Except in the triangle case, it need not be equiangular (need not have all angles equal), but if it does then it is a regular polygon . If the number of sides is at least five, an equilateral polygon need not be a convex polygon : it could be concave or even self-intersecting . CONTENTS * 1 Examples * 2 Triambi * 3 References * 4 External links EXAMPLESAll regular polygons and isotoxal polygons are equilateral. An equilateral triangle is a regular triangle with 60° internal angles . An equilateral quadrilateral is called a rhombus , an isotoxal polygon described by an angle α. It includes the square as a special case. A convex equilateral pentagon can be described by two angles α and β, which together determine the other angles
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Isogonal Figure
In geometry , a polytope (a polygon , polyhedron or tiling, for example) is ISOGONAL or VERTEX-TRANSITIVE if, loosely speaking, all its vertices are equivalent. That implies that