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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. CONTENTS * 1 Description * 2 Cases * 2.1 Symmetry groups * 2.2 Regular polygons (plane) * 2.3 Regular polyhedra (3 dimensions) * 2.4 Regular 4-polytopes (4 dimensions) * 2.5 Regular n-polytopes (higher dimensions) * 2.6 Dual polytopes * 2.7 Prismatic polytopes * 3 Extension of Schläfli symbols * 3.1 Polygons and circle tilings * 3.2 Polyhedra and tilings * 3.2.1 Alternations, quarters and snubs * 3.2.2 Altered and holosnubbed * 3.3 Polychora and honeycombs * 3.3.1 Alternations, quarters and snubs * 3.3.2 Bifurcating families * 4 See also * 5 References * 6 Sources * 7 External links DESCRIPTIONThe Schläfli symbol
Schläfli symbol
is a recursive description, starting with {p} for a p-sided regular polygon that is convex . For example, {3} is an equilateral triangle , {4} is a square , {5} a convex regular pentagon and so on. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices. For example, {5/2} is a pentagram
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Dodecahedron
In geometry , a DODECAHEDRON (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron , which is a Platonic solid
Platonic solid
. There are also three regular star dodecahedra , which are constructed as stellations of the convex form. All of these have icosahedral symmetry , order 120. The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry . The rhombic dodecahedron , seen as a limiting case of the pyritohedron, has octahedral symmetry . The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling . There are a large number of other dodecahedra
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Pentagon
In geometry , a PENTAGON (from the Greek πέντε pente and γωνία gonia, meaning five and angle ) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting . A self-intersecting regular pentagon (or star pentagon) is called a pentagram . CONTENTS* 1 Regular pentagons * 1.1 Derivation of the area formula * 1.2 Inradius * 1.3 Chords from the circumscribed circle to the vertices * 1.4 Construction of a regular pentagon * 1.4.1 Richmond\'s method * 1.4.2 Carlyle circles * 1.4.3 Using trigonometry and the Pythagorean Theorem * 1.4.3.1 The construction * 1.4.4 † Proof that cos 36° = 1 + 5 4 {displaystyle {tfrac {1+{sqrt {5}}}{4}}} * 1.4.5 Side length is given * 1.4.5.1 The golden ratio * 1.5 Euclid\'s method * 1.5.1 Simply using a protractor (not a classical construction) * 1.6 Physical methods * 1.7 Symmetry * 2 Equilateral pentagons * 3 Cyclic pentagons * 4 General convex pentagons * 5 Graphs * 6 Examples of pentagons * 6.1 Plants * 6.2 Animals * 6.3 Artificial * 7 Pentagons in tiling * 8 Pentagons in polyhedra * 9 See also * 10 In-line notes and references * 11 External links REGULAR PENTAGONSA regular pentagon has Schläfli symbol
Schläfli symbol
{5} and interior angles are 108°
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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. OF A POLYTOPEA vertex is a corner point of a polygon , polyhedron , or other higher-dimensional polytope , formed by the intersection of edges , faces or facets of the object. In a polygon, a vertex is called "convex " if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians ( 180°, two right angles ) ; otherwise, it is called "concave" or "reflex"
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; _geo-_ "earth", _-metron_ "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer . Geometry arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages . By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat . Since then, and into modern times, geometry has expanded into non- Euclidean geometry and manifolds , describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry
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List Of Regular Polytopes
This page lists the regular polytopes and regular polytope compounds in Euclidean , spherical and hyperbolic spaces. The Schläfli symbol
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
Coxeter group
, which Coxeter
Coxeter
expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter
Coxeter
notation . Another related symbol is the Coxeter-Dynkin diagram
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 {4,3}, and with its octahedral symmetry , or , it is represented by Coxeter
Coxeter
diagram . The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets . Infinite forms tessellate a one-lower-dimensional Euclidean space
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Ludwig Schläfli
LUDWIG SCHLäFLI (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces. The concept of multidimensionality has come to play a pivotal role in physics , and is a common element in science fiction. CONTENTS* 1 Life and career * 1.1 Youth and education * 1.2 Teaching * 1.3 Later life * 2 Higher dimensions * 3 Polytopes * 4 See also * 5 References * 6 External links LIFE AND CAREERYOUTH AND EDUCATIONLudwig spent most of his life in Switzerland
Switzerland
. He was born in Grasswil (now part of Seeberg ), his mother's hometown. The family then moved to the nearby Burgdorf , where his father worked as a tradesman . His father wanted Ludwig to follow in his footsteps, but Ludwig was not cut out for practical work. In contrast, because of his mathematical gifts, he was allowed to attend the Gymnasium in Bern
Bern
in 1829. By that time he was already learning differential calculus from Abraham Gotthelf Kästner 's Mathematische Anfangsgründe der Analysis des Unendlichen (1761). In 1831 he transferred to the Akademie in Bern
Bern
for further studies. By 1834 the Akademie had become the new Universität Bern
Bern
, where he started studying theology
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Recursive Definition
A RECURSIVE DEFINITION (or INDUCTIVE DEFINITION) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff). A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs. For example, the factorial function n! is defined by the rules 0! = 1. (n+1)! = (n+1)·n!. This definition is valid for all n, because the recursion eventually reaches the BASE CASE of 0. The definition may also be thought of as giving a procedure describing how to construct the function n!, starting from n = 0 and proceeding onwards with n = 1, n = 2, n = 3 etc.. The recursion theorem states that such a definition indeed defines a function. The proof uses mathematical induction . An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set N of natural numbers is: * 1 is in N. * If an element n is in N then n+1 is in N. * N is the intersection of all sets satisfying (1) and (2).There are many sets that satisfy (1) and (2) - for example, the set {1, 1.649, 2, 2.649, 3, 3.649, ...} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members
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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
Area
(with _s_=side length) A = 1 4 n s 2 cot n {displaystyle A={tfrac {1}{4}}ns^{2}cot {frac {pi }{n}}} Internal angle ( n 2 ) 180 n {displaystyle (n-2)times {frac {180^{circ }}{n}}} Internal angle sum ( n 2 ) 180 {displaystyle left(n-2right)times 180^{circ }} Inscribed circle diameter d I C = s cot n {displaystyle d_{IC}=scot {frac {pi }{n}}} Circumscribed circle diameter d O C = s csc n {displaystyle d_{OC}=scsc {frac {pi }{n}}} Properties convex , cyclic , equilateral , isogonal , isotoxal In Euclidean geometry , a REGULAR POLYGON is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be CONVEX or STAR . In the limit , a sequence of regular polygons with an increasing number of sides approximates a circle , if the perimeter is fixed, or a regular apeirogon , if the edge length is fixed
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Convex Set
In convex geometry , a CONVEX SET is a subset of an affine space that is closed under convex combinations . More specifically, in a Euclidean space
Euclidean space
, a CONVEX REGION is a region where, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve . The intersection of all convex sets containing a given subset A of Euclidean space
Euclidean space
is called the convex hull of A. It is the smallest convex set containing A. A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis . The notion of a convex set can be generalized as described below
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Equilateral Triangle
In geometry , an EQUILATERAL TRIANGLE is a triangle in which all three sides are equal. In the familiar Euclidean geometry
Euclidean geometry
, equilateral triangles are also equiangular ; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons , and can therefore also be referred to as regular triangles. CONTENTS * 1 Principal properties * 2 Characterizations * 2.1 Sides * 2.2 Semiperimeter
Semiperimeter
* 2.3 Angles * 2.4 Area
Area
* 2.5 Circumradius, inradius and exradii * 2.6 Equal cevians * 2.7 Coincident triangle centers * 2.8 Six triangles formed by partitioning by the medians * 2.9 Points in the plane * 3 Notable theorems * 4 Other properties * 5 Geometric construction * 6 Derivation of area formula * 6.1 Using the Pythagorean theorem
Pythagorean theorem
* 6.2 Using trigonometry * 7 In culture and society * 8 See also * 9 References * 10 External links PRINCIPAL PROPERTIES An equilateral triangle. It has equal sides (a=b=c), equal angles ( = = {displaystyle alpha =beta =gamma } ), and equal altitudes (ha=hb=hc)
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Square (geometry)
In geometry , a SQUARE is a regular quadrilateral , which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles ). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted {displaystyle square } ABCD. CONTENTS * 1 Characterizations * 2 Properties * 2.1 Perimeter
Perimeter
and area * 2.2 Other facts * 3 Coordinates and equations * 4 Construction * 5 Symmetry * 6 Squares inscribed in triangles * 7 Squaring the circle
Squaring the circle
* 8 Non-Euclidean geometry * 9 Crossed square * 10 Graphs * 11 See also * 12 References * 13 External links CHARACTERIZATIONSA convex quadrilateral is a square if and only if it is any one of the following: * a rectangle with two adjacent equal sides * a rhombus with a right vertex angle * a rhombus with all angles equal * a parallelogram with one right vertex angle and two adjacent equal sides * a quadrilateral with four equal sides and four right angles * a quadrilateral where the diagonals are equal and are the perpendicular bisectors of each other, i.e. a rhombus with equal diagonals * a convex quadrilateral with successive sides a, b, c, d whose area is A = 1 2 ( a 2 + c 2 ) = 1 2 ( b 2 + d 2 )
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Regular Pentagon
In geometry , a PENTAGON (from the Greek πέντε pente and γωνία gonia, meaning five and angle ) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting . A self-intersecting regular pentagon (or star pentagon) is called a pentagram . CONTENTS* 1 Regular pentagons * 1.1 Derivation of the area formula * 1.2 Inradius * 1.3 Chords from the circumscribed circle to the vertices * 1.4 Construction of a regular pentagon * 1.4.1 Richmond\'s method * 1.4.2 Carlyle circles * 1.4.3 Using trigonometry and the Pythagorean Theorem * 1.4.3.1 The construction * 1.4.4 † Proof that cos 36° = 1 + 5 4 {displaystyle {tfrac {1+{sqrt {5}}}{4}}} * 1.4.5 Side length is given * 1.4.5.1 The golden ratio * 1.5 Euclid\'s method * 1.5.1 Simply using a protractor (not a classical construction) * 1.6 Physical methods * 1.7 Symmetry * 2 Equilateral pentagons * 3 Cyclic pentagons * 4 General convex pentagons * 5 Graphs * 6 Examples of pentagons * 6.1 Plants * 6.2 Animals * 6.3 Artificial * 7 Pentagons in tiling * 8 Pentagons in polyhedra * 9 See also * 10 In-line notes and references * 11 External links REGULAR PENTAGONSA regular pentagon has Schläfli symbol
Schläfli symbol
{5} and interior angles are 108°
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Star Polygon
In geometry , a STAR POLYGON is a type of non-convex polygon . Only the REGULAR STAR POLYGONS have been studied in any depth; star polygons in general appear not to have been formally defined. Branko Grünbaum
Branko Grünbaum
identified two primary definitions used by Kepler
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 . CONTENTS * 1 Etymology * 2 Regular star polygon * 2.1 Degenerate regular star polygons * 3 Simple isotoxal star polygons * 4 Interiors of star polygons * 5 Star polygons in art and culture * 6 See also * 7 References ETYMOLOGY Star polygon
Star polygon
names combine a numeral prefix , such as penta-, with the Greek suffix -gram (in this case generating the word pentagram ). The prefix is normally a Greek cardinal , but synonyms using other prefixes exist. For example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin . The -gram suffix derives from γραμμή (grammḗ) meaning a line. REGULAR STAR POLYGON Further information: Regular polygon
Regular polygon
§ Regular star polygons {5/2} {7/2} {7/3} ..
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Irreducible Fraction
An IRREDUCIBLE FRACTION (or FRACTION IN LOWEST TERMS or REDUCED FRACTION) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction a⁄b is irreducible if and only if a and b are coprime , that is, if a and b have a greatest common divisor of 1. In higher mathematics , "IRREDUCIBLE FRACTION" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials . Every positive rational number can be represented as an irreducible fraction in exactly one way. An equivalent definition is sometimes useful: if a, b are integers, then the fraction a⁄b is irreducible if and only if there is no other equal fraction c⁄d such that c < a or d < b, where a means the absolute value of a. (Two fractions a⁄b and c⁄d are equal or equivalent if and only if ad = bc.) For example, 1⁄4, 5⁄6, and −101⁄100 are all irreducible fractions. On the other hand, 2⁄4 is reducible since it is equal in value to 1⁄2, and the numerator of 1⁄2 is less than the numerator of 2⁄4. A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor . In order to find the greatest common divisor, the Euclidean algorithm or prime factorization may be used
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Pentagram
A PENTAGRAM (sometimes known as a PENTALPHA or PENTANGLE or a STAR PENTAGON ) is the shape of a five-pointed star drawn with five straight strokes. Pentagrams were used symbolically in ancient Greece and Babylonia
Babylonia
, and are used today as a symbol of faith by many Wiccans , akin to the use of the cross by Christians and the Star of David
Star of David
by Jews. The pentagram has magical associations, and many people who practice Neopagan faiths wear jewelry incorporating the symbol. Christians once more commonly used the pentagram to represent the five wounds of Jesus . The pentagram has associations with Freemasonry
Freemasonry
and is also utilized by other belief systems. The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line". The word "pentacle " is sometimes used synonymously with "pentagram" The word pentalpha is a learned modern (17th-century) revival of a post-classical Greek name of the shape
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