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In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 …

Symmetries of a regular icosagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.

Image: 4.5.20 vertex

Image: 10 10 duopyramid ortho 3

Image: 10 10 duoprism ortho 3

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1. Icosagon – In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagons interior angles is 3240 degrees, the regular icosagon has Schläfli symbol, and can also be constructed as a truncated decagon, t, or a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°, the area of a regular icosagon with edge length t is A =5 t 2 ≃31.5687 t 2. In terms of the radius R of its circumcircle, the area is A =5 R22, since the area of the circle is π R2, the Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section. The Globe, the outdoor theater used by William Shakespeares acting company, was discovered to have built on an icosagonal foundation when a partial excavation was done in 1989. As a golygonal path, the swastika is considered to be an irregular icosagon, a regular square, pentagon, and icosagon can completely fill a plane vertex. E20 E1 ¯ E1 F ¯ = E20 F ¯ E20 E1 ¯ =1 +52 = φ ≈1.618 The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries, and, and 6 cyclic group symmetries and these 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order, full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g20 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular icosagon. An icosagram is a 20-sided star polygon, represented by symbol, there are three regular forms given by Schläfli symbols, and. There are also five regular star figures using the vertex arrangement,2,4,5,2,4. Deeper truncations of the regular decagon and decagram can produce isogonal intermediate icosagram forms with equally spaced vertices, a regular icosagram, can be seen as a quasitruncated decagon, t=. Similarly a decagram, has a quasitruncation t=, and finally a simple truncation of a decagram gives t=

2. Geometry – Geometry 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 mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. 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 footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

3. Pentagon – In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting, a self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°, a regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan 4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan 2 =5 t 2 tan 4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2

4. The Price Is Right (U.S. game show) – The Price Is Right is an American television game show created by Bob Stewart, Mark Goodson and Bill Todman. The show revolves around contestants competing to identify accurate pricing of merchandise to win cash, Contestants are selected from the studio audience when the announcer proclaims the shows famous catchphrase, Come on down. The program premiered on September 4,1972 on CBS, Bob Barker was the series longest-running host from its 1972 debut until his retirement in June 2007, when Drew Carey took over. Barker was accompanied by a series of announcers, beginning with Johnny Olson, followed by Rod Roddy, in April 2011, George Gray became the announcer. The show has used models, most notably Anitra Ford, Janice Pennington, Dian Parkinson, Holly Hallstrom. While retaining some elements of the version of the show. The Price Is Right has aired over 8,000 episodes since its debut and is one of the network series in United States television history. In a 2007 article, TV Guide named The Price Is Right the greatest game show of all time, the shows 45th season premiered on September 19,2016. At the beginning of the show, four contestants are called from the audience by the announcer to take a spot on the front row behind bidding podiums and this area is known as Contestants Row. The announcer shouts Come on down, after calling each selected contestants name, a phrase which has become a trademark of the show. The four contestants in Contestants Row compete in a round to determine which contestant will play the next pricing game. A prize is shown and each contestant gives a single bid for the item, in the first One-Bid game of each episode, bidding begins with the contestant on the viewers left-to-right. In subsequent One-Bid rounds, the order of bidding moves from the viewers left-to-right. Contestants are instructed to bid in whole dollars since the price of the item is rounded to the nearest dollar. If a contestant thinks the others have overbid, he or she bids $1 on the item, the contestant whose bid is closest to the actual retail price of the prize without going over wins that prize and gets to play the subsequent pricing game. If all four contestants overbid, short buzzer tones sound, the lowest bid is announced, the host then instructs the contestants to re-bid below the lowest previous bid. If a contestant bids the actual retail price, a bell rings, from the introduction of the bonus in 1977 until 1998, the perfect bid bonus was $100, it was permanently increased to the current $500 in 1998. On The Price Is Right $1,000,000 Spectacular, after each pricing game, another contestant is called to come on down to fill the spot of the contestant that won the previous round