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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.
Chirality
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Chirality /kaɪˈrælɪtiː/ is a property of asymmetry important in several branches of science. The word chirality is derived from the Greek, χειρ, hand, an object or a system is chiral if it is distinguishable from its mirror image, that is, it cannot be superposed onto it. Conversely, an image of an achiral object, such as a sphere. A chiral object and its image are called enantiomorphs or. A non-chiral object is called achiral and can be superposed on its mirror image, human hands are perhaps the most universally recognized example of chirality. The left hand is a mirror image of the right hand. This difference in symmetry becomes obvious if someone attempts to shake the hand of a person using their left hand. In mathematics, chirality is the property of a figure that is not identical to its mirror image, in mathematics, a figure is chiral if it cannot be mapped to its mirror image by rotations and translations alone. For example, a shoe is different from a left shoe. A chiral object and its image are said to be enantiomorphs. The word enantiomorph stems from the Greek ἐναντίος opposite + μορφή form, a non-chiral figure is called achiral or amphichiral. The helix and Möbius strip are chiral two dimensional objects in three dimensional ambient space, the J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two dimensional space. Many other familiar objects exhibit the same symmetry of the human body, such as gloves, glasses. A similar notion of chirality is considered in theory, as explained below. Some chiral three dimensional objects, such as the helix, can be assigned a right or left handedness, in geometry a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In two dimensions, every figure that possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry, in three dimensions, every figure that possesses a plane of symmetry or a center of symmetry is achiral. There are, however, achiral figures lacking both plane and center of symmetry, in terms of point groups, all chiral figures lack an improper axis of rotation. This means that they contain a center of inversion or a mirror plane

3.
Regular map (graph theory)
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In mathematics, a regular map is a symmetric tessellation of a closed surface. Regular maps are, in a sense, topological generalizations of Platonic solids, the theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory. Regular maps are classified according to either, the genus and orientability of the supporting surface, regular maps are typically defined and studied in three ways, topologically, group-theoretically, and graph-theoretically. Topologically, a map is a 2-cell decomposition of a closed compact 2-manifold and it is a crucial fact that there is a finite number of regular maps for every orientable genus except the torus. In this definition the faces are the orbits of F = <r0, r1>, edges are the orbits of E = <r0, r2>, more abstractly, the automorphism group of any regular map is the non-degenerate, homomorphic image of a <2, m, n>-triangle group. Note that Γ is the graph or graph encoded map of the map. In general, | Ω | = 4|E|, a map M is regular iff Aut acts regularly on the flags. Aut of a map is transitive on the vertices, edges. A map M is said to be reflexible iff Aut is regular, a map which is regular but not reflexible is said to be chiral. The great dodecahedron is a map with pentagonal faces in the orientable surface of genus 4. The hemicube is a map of type in the projective plane. The hemi-dodecahedron is a map produced by pentagonal embedding of the Petersen graph in the projective plane. The p-hosohedron is a map of type. Note that the hosohedra are non-polyhedral in the sense that they are not abstract polytopes, in particular, they do not satisfy the diamond property. The Dyck map is a map of 12 octagons on a genus-3 surface. Its underlying graph, the Dyck graph, can form a regular map of 16 hexagons in a torus. The following is a complete list of maps in surfaces of positive Euler characteristic, χ, the sphere. The images below show three of the 20 regular maps in the torus, labelled with their Schläfli symbols