1.
Octagon
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In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can also be constructed as a truncated square, t. A truncated octagon, t is a hexadecagon, t, the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°, the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. A regular octagon is a figure with sides of the same length. It has eight lines of symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol, the internal angle at each vertex of a regular octagon is 135°. The area of an octagon of side length a is given by A =2 cot π8 a 2 =2 a 2 ≃4.828 a 2. In terms of the circumradius R, the area is A =4 sin π4 R2 =22 R2 ≃2.828 R2. In terms of the r, the area is A =8 tan π8 r 2 =8 r 2 ≃3.314 r 2. These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as A = S2 − a 2, where S is the span of the octagon, or the second-shortest diagonal, and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside and then takes the corner triangles and places them with right angles pointed inward, the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈2.414 a. The area is then as above, A =2 − a 2 =2 a 2 ≈4.828 a 2, expressed in terms of the span, the area is A =2 S2 ≈0.828 S2. Another simple formula for the area is A =2 a S, more often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S /2.414, the two end lengths e on each side, as well as being e = a /2, may be calculated as e = /2. The circumradius of the octagon in terms of the side length a is R = a
2.
Geometry
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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.
Oxford English Dictionary
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The Oxford English Dictionary is a descriptive dictionary of the English language, published by the Oxford University Press. The second edition came to 21,728 pages in 20 volumes, in 1895, the title The Oxford English Dictionary was first used unofficially on the covers of the series, and in 1928 the full dictionary was republished in ten bound volumes. In 1933, the title The Oxford English Dictionary fully replaced the name in all occurrences in its reprinting as twelve volumes with a one-volume supplement. More supplements came over the years until 1989, when the edition was published. Since 2000, an edition of the dictionary has been underway. The first electronic version of the dictionary was available in 1988. The online version has been available since 2000, and as of April 2014 was receiving two million hits per month. The third edition of the dictionary will probably appear in electronic form, Nigel Portwood, chief executive of Oxford University Press. As a historical dictionary, the Oxford English Dictionary explains words by showing their development rather than merely their present-day usages, therefore, it shows definitions in the order that the sense of the word began being used, including word meanings which are no longer used. The format of the OEDs entries has influenced numerous other historical lexicography projects and this influenced later volumes of this and other lexicographical works. As of 30 November 2005, the Oxford English Dictionary contained approximately 301,100 main entries, the dictionarys latest, complete print edition was printed in 20 volumes, comprising 291,500 entries in 21,730 pages. The longest entry in the OED2 was for the verb set, as entries began to be revised for the OED3 in sequence starting from M, the longest entry became make in 2000, then put in 2007, then run in 2011. Despite its impressive size, the OED is neither the worlds largest nor the earliest exhaustive dictionary of a language, the Dutch dictionary Woordenboek der Nederlandsche Taal is the worlds largest dictionary, has similar aims to the OED and took twice as long to complete. Another earlier large dictionary is the Grimm brothers dictionary of the German language, begun in 1838, the official dictionary of Spanish is the Diccionario de la lengua española, and its first edition was published in 1780. The Kangxi dictionary of Chinese was published in 1716, trench suggested that a new, truly comprehensive dictionary was needed. On 7 January 1858, the Society formally adopted the idea of a new dictionary. Volunteer readers would be assigned particular books, copying passages illustrating word usage onto quotation slips, later the same year, the Society agreed to the project in principle, with the title A New English Dictionary on Historical Principles. He withdrew and Herbert Coleridge became the first editor, on 12 May 1860, Coleridges dictionary plan was published and research was started
4.
Regular polytope
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike, note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form, with regular facets as, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object, some of these have regular examples, as discussed in the section on historical discovery below. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, the notation is best explained by adding one dimension at a time. A convex regular polygon having n sides is denoted by, so an equilateral triangle is, a square, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the figure of the polyhedron. A regular 4-polytope having cells with q cells joining around an edge is denoted by, the vertex figure of the 4-polytope is a. A five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to
5.
Network topology
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Network topology is the arrangement of the various elements of a computer network. Essentially, it is the structure of a network and may be depicted physically or logically. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two networks, yet their topologies may be identical, an example is a local area network. Conversely, mapping the data flow between the components determines the topology of the network. Two basic categories of network topologies exist, physical topologies and logical topologies, the cabling layout used to link devices is the physical topology of the network. This refers to the layout of cabling, the locations of nodes, a networks logical topology is not necessarily the same as its physical topology. For example, the twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token ring is a ring topology, but is wired as a physical star from the media access unit. Logical topologies are often associated with media access control methods. Some networks are able to change their logical topology through configuration changes to their routers. The study of network topology recognizes eight basic topologies, point-to-point, bus, star, ring or circular, mesh, tree, hybrid, the simplest topology with a dedicated link between two endpoints. Easiest to understand, of the variations of point-to-point topology, is a point-to-point communications channel that appears, to the user, a childs tin can telephone is one example of a physical dedicated channel. Using circuit-switching or packet-switching technologies, a point-to-point circuit can be set up dynamically, switched point-to-point topologies are the basic model of conventional telephony. The value of a permanent point-to-point network is unimpeded communications between the two endpoints, the value of an on-demand point-to-point connection is proportional to the number of potential pairs of subscribers and has been expressed as Metcalfes Law. In local area networks where bus topology is used, each node is connected to a single cable and this central cable is the backbone of the network and is known as the bus. A signal from the travels in both directions to all machines connected on the bus cable until it finds the intended recipient. If the machine address does not match the address for the data. Alternatively, if the matches the machine address, the data is accepted
6.
Parallel computing
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Parallel computing is a type of computation in which many calculations or the execution of processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time, there are several different forms of parallel computing, bit-level, instruction-level, data, and task parallelism. Parallelism has been employed for years, mainly in high-performance computing. As power consumption by computers has become a concern in recent years, parallel computing has become the dominant paradigm in computer architecture, specialized parallel computer architectures are sometimes used alongside traditional processors, for accelerating specific tasks. Communication and synchronization between the different subtasks are typically some of the greatest obstacles to getting good parallel program performance, a theoretical upper bound on the speed-up of a single program as a result of parallelization is given by Amdahls law. Traditionally, computer software has been written for serial computation, to solve a problem, an algorithm is constructed and implemented as a serial stream of instructions. These instructions are executed on a central processing unit on one computer, only one instruction may execute at a time—after that instruction is finished, the next one is executed. Parallel computing, on the hand, uses multiple processing elements simultaneously to solve a problem. This is accomplished by breaking the problem into independent parts so that each processing element can execute its part of the algorithm simultaneously with the others. The processing elements can be diverse and include such as a single computer with multiple processors, several networked computers, specialized hardware. Frequency scaling was the dominant reason for improvements in performance from the mid-1980s until 2004. The runtime of a program is equal to the number of instructions multiplied by the time per instruction. Maintaining everything else constant, increasing the frequency decreases the average time it takes to execute an instruction. An increase in frequency thus decreases runtime for all compute-bound programs. However, power consumption P by a chip is given by the equation P = C × V2 × F, where C is the capacitance being switched per clock cycle, V is voltage, increases in frequency increase the amount of power used in a processor. Moores law is the observation that the number of transistors in a microprocessor doubles every 18 to 24 months. Despite power consumption issues, and repeated predictions of its end, with the end of frequency scaling, these additional transistors can be used to add extra hardware for parallel computing. Optimally, the speedup from parallelization would be linear—doubling the number of processing elements should halve the runtime, however, very few parallel algorithms achieve optimal speedup
7.
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
8.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
9.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
10.
Charles Howard Hinton
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Charles Howard Hinton was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension and he is known for coining the word tesseract and for his work on methods of visualising the geometry of higher dimensions. Hinton taught at Cheltenham College while he studied at Balliol College, Oxford, from 1880 to 1886, he taught at Uppingham School in Rutland, where Howard Candler, a friend of Edwin Abbott Abbotts, also taught. Hinton also received his M. A. from Oxford in 1886, in 1880 Hinton married Mary Ellen, daughter of Mary Everest Boole and George Boole, the founder of mathematical logic. The couple had four children, George, Eric, William, in 1883 he went through a marriage ceremony with Maud Florence, by whom he had had twin children, under the assumed identity of John Weldon. He was subsequently convicted of bigamy and spent three days in prison, losing his job at Uppingham, in 1887 Charles moved with Mary Ellen to Japan to work in a mission before accepting a job as headmaster of the Victoria Public School. In 1893 he sailed to the United States on the SS Tacoma to take up a post at Princeton University as an instructor in mathematics, in 1897, he designed a gunpowder-powered baseball pitching machine for the Princeton baseball teams batting practice. The machine was versatile, capable of variable speeds with an adjustable breech size, at the end of his life, Hinton worked as an examiner of chemical patents for the United States Patent Office. At age 54, he died unexpectedly of a hemorrhage on 30 April 1907. After Hintons sudden death his wife, Mary Ellen, committed suicide in Washington, in an 1880 article entitled What is the Fourth Dimension. Hinton calls the casting out the self, equates it with the process of sympathizing with another person. Hinton created several new words to describe elements in the fourth dimension, according to OED, he first used the word tesseract in 1888 in his book A New Era of Thought. He also invented the words kata and ana to describe the two opposing fourth-dimensional directions. Hintons Scientific romances, including What is the Fourth Dimension. and A Plane World, were published as a series of nine pamphlets by Swan Sonnenschein & Co. during 1884–1886. In the introduction to A Plane World, Hinton referred to Abbotts recent Flatland as having similar design, Abbott used the stories as a setting wherein to place his satire and his lessons. But we wish in the first place to know the physical facts, Hintons world existed along the perimeter of a circle rather than on an infinite flat plane. He extended the connection to Abbotts work with An Episode on Flatland, Hinton was one of the many thinkers who circulated in Jorge Luis Borgess pantheon of writers. Hinton is mentioned in Borges short stories Tlön, Uqbar, Orbis Tertius, There Are More Things and El milagro secreto, many of ideas Ouspensky presents in Tertium Organum mention Hintons works
11.
Rotations in 4-dimensional Euclidean space
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In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO. The name comes from the fact that it is the orthogonal group of order 4. In this article rotation means rotational displacement, for the sake of uniqueness rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise. A fixed plane is a plane for which every vector in the plane is unchanged after the rotation, an invariant plane is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation. Four-dimensional rotations are of two types, simple rotations and double rotations, a simple rotation R about a rotation centre O leaves an entire plane A through O fixed. Every plane B that is orthogonal to A intersects A in a certain point P. Each such point P is the centre of the 2D rotation induced by R in B, all these 2D rotations have the same rotation angle α. Half-lines from O in the axis-plane A are not displaced, half-lines from O orthogonal to A are displaced through α, all other half-lines are displaced through an angle < α. For each rotation R of 4-space, there is at least one pair of orthogonal 2-planes A and B each of which are invariant, hence R operating on either of these planes produces an ordinary rotation of that plane. For almost all R, the rotation angles α in plane A and β in plane B — both assumed to be nonzero — are different, the unequal rotation angles α and β satisfying -π < α, β < π are almost* uniquely determined by R. Assuming that 4-space is oriented, then the orientations of the 2-planes A and B can be consistent with this orientation in two ways. If the rotation angles are unequal, R is sometimes termed a double rotation, *Assuming that 4-space is oriented, then an orientation for each of the 2-planes A and B can be chosen to be consistent with this orientation of 4-space in two equally valid ways. If the angles from one choice of orientations of A and B are. If the rotation angles of a rotation are equal then there are infinitely many invariant planes instead of just two, and all half-lines from O are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements, beware, not all planes through O are invariant under isoclinic rotations, only planes that are spanned by a half-line and the corresponding displaced half-line are invariant. Assuming that an orientation has been chosen for 4-dimensional space. Now assume that only the rotation angle α is specified, then there are in general four isoclinic rotations in planes OUX and OYZ with rotation angle α, depending on the rotation senses in OUX and OYZ. We make the convention that the senses from OU to OX
12.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face