1.
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.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a 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 a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a 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. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
3.
Bell Labs
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Nokia Bell Labs is an American research and scientific development company, owned by Finnish company Nokia. Its headquarters are located in Murray Hill, New Jersey, in addition to laboratories around the rest of the United States. The historic laboratory originated in the late 19th century as the Volta Laboratory, Bell Labs was also at one time a division of the American Telephone & Telegraph Company, half-owned through its Western Electric manufacturing subsidiary. Eight Nobel Prizes have been awarded for work completed at Bell Laboratories, in 1880, the French government awarded Alexander Graham Bell the Volta Prize of 50,000 francs, approximately US$10,000 at that time for the invention of the telephone. Bell used the award to fund the Volta Laboratory in Washington, D. C. in collaboration with Sumner Tainter, the laboratory is also variously known as the Volta Bureau, the Bell Carriage House, the Bell Laboratory and the Volta Laboratory. The laboratory focused on the analysis, recording, and transmission of sound, Bell used his considerable profits from the laboratory for further research and education to permit the diffusion of knowledge relating to the deaf. This resulted in the founding of the Volta Bureau c,1887, located at Bells fathers house at 1527 35th Street in Washington, D. C. where its carriage house became their headquarters in 1889. In 1893, Bell constructed a new building, close by at 1537 35th St. specifically to house the lab, the building was declared a National Historic Landmark in 1972. In 1884, the American Bell Telephone Company created the Mechanical Department from the Electrical, the first president of research was Frank B. Jewett, who stayed there until 1940, ownership of Bell Laboratories was evenly split between AT&T and the Western Electric Company. Its principal work was to plan, design, and support the equipment that Western Electric built for Bell System operating companies and this included everything from telephones, telephone exchange switches, and transmission equipment. Bell Labs also carried out consulting work for the Bell Telephone Company, a few workers were assigned to basic research, and this attracted much attention, especially since they produced several Nobel Prize winners. Until the 1940s, the principal locations were in and around the Bell Labs Building in New York City. Of these, Murray Hill and Crawford Hill remain in existence, the largest grouping of people in the company was in Illinois, at Naperville-Lisle, in the Chicago area, which had the largest concentration of employees prior to 2001. Since 2001, many of the locations have been scaled down or closed. The Holmdel site, a 1.9 million square foot structure set on 473 acres, was closed in 2007, the mirrored-glass building was designed by Eero Saarinen. In August 2013, Somerset Development bought the building, intending to redevelop it into a commercial and residential project. The prospects of success are clouded by the difficulty of readapting Saarinens design and by the current glut of aging, eight Nobel Prizes have been awarded for work completed at Bell Laboratories
4.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
5.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
6.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
7.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
8.
Icosidodecahedron
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In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly and its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, the icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae. In this form its symmetry is D5d, order 20, the wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the permutations of. The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a face. The last two correspond to the A2 and H2 Coxeter planes, the icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The icosidodecahedron is a dodecahedron and also a rectified icosahedron. With orbifold notation symmetry of all of these tilings are wythoff construction within a fundamental domain of symmetry. The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images, the icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves. Eight uniform star polyhedra share the same vertex arrangement, of these, two also share the same edge arrangement, the small icosihemidodecahedron, and the small dodecahemidodecahedron. The vertex arrangement is shared with the compounds of five octahedra. In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words, the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons, six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron, in the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids
9.
Connectivity (graph theory)
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It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network, a graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices, a graph that is not connected is disconnected. A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints, a graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected, in an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are connected by a path of length 1, i. e. by a single edge. A graph is said to be connected if every pair of vertices in the graph is connected, a connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge, a directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected graph. It is connected if it contains a path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected, diconnected, or simply strong if it contains a path from u to v. The strong components are the maximal strongly connected subgraphs, a cut, vertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. The connectivity or vertex connectivity κ is the size of a minimal vertex cut, a graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater. In particular, a graph with n vertices, denoted Kn, has no vertex cuts at all. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. The local connectivity κ is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs, a graph G which is connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges, in the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, a cut of G is a set of edges whose removal renders the graph disconnected. A graph is called k-edge-connected if its edge connectivity is k or greater, if u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex
10.
Complete bipartite graph
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Graph theory itself is typically dated as beginning with Leonhard Eulers 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, Llull himself had made similar drawings of complete graphs three centuries earlier. That is, it is a graph such that for every two vertices v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km, n, for any k, K1, k is called a star. All complete bipartite graphs which are trees are stars, the graph K1,3 is called a claw, and is used to define the claw-free graphs. The graph K3,3 is called the utility graph and this usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings, it is impossible to solve without crossings due to the nonplanarity of K3,3. Given a bipartite graph, testing whether it contains a complete bipartite subgraph Ki, a planar graph cannot contain K3,3 as a minor, an outerplanar graph cannot contain K3,2 as a minor. Conversely, every nonplanar graph contains either K3,3 or the complete graph K5 as a minor, Kn, n is a Moore graph and a -cage. The complete bipartite graphs Kn, n and Kn, n+1 have the possible number of edges among all triangle-free graphs with the same number of vertices. The complete bipartite graph Km, n has a vertex covering number of min, the complete bipartite graph Km, n has a maximum independent set of size max. The adjacency matrix of a bipartite graph Km, n has eigenvalues √, −√ and 0, with multiplicity 1,1. The Laplacian matrix of a bipartite graph Km, n has eigenvalues n+m, n, m. A complete bipartite graph Km, n has mn−1 nm−1 spanning trees, a complete bipartite graph Km, n has a maximum matching of size min. A complete bipartite graph Kn, n has a proper n-edge-coloring corresponding to a Latin square, every complete bipartite graph is a modular graph, every triple of vertices has a median that belongs to shortest paths between each pair of vertices
11.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio