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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Dodecagon
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In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=

3.
Dihedral group
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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 groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3