1.
Perspective (graphical)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the spot as the windowpane was painted. Each painted object in the scene is thus a flat, scaled down version of the object on the side of the window. All perspective drawings assume the viewer is a distance away from the drawing. Objects are scaled relative to that viewer, an object is often not scaled evenly, a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening, Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewers eye, represents objects infinitely far away and they have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earths horizon, any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a vanishing point, usually directly opposite the viewers eye. All lines parallel with the line of sight recede to the horizon towards this vanishing point. This is the standard receding railroad tracks phenomenon, a two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of lines that are at an angle relative to the plane of the drawing. Perspectives consisting of parallel lines are observed most often when drawing architecture. In contrast, natural scenes often do not have any sets of parallel lines, the only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles. Chinese artists made use of perspective from the first or second century until the 18th century. It is not certain how they came to use the technique, some authorities suggest that the Chinese acquired the technique from India, oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga. This was detailed within Aristotles Poetics as skenographia, using flat panels on a stage to give the illusion of depth, the philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage, Euclids Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclids perspective coincides with the modern mathematical definition
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.
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
4.
Connection Machine
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Danny Hillis and Sheryl Handler founded Thinking Machines Corporation in Waltham, Massachusetts in 1983, moving in 1984 to Cambridge, MA. The CM-1, depending on the configuration, has as many as 65,536 individual processors, each extremely simple, CM-1 and its successor CM-2 take the form of a cube 1.5 meters on a side, divided equally into eight smaller cubes. Each subcube contains 16 printed circuit boards and a main processor called a sequencer, each circuit board contains 32 chips. Each chip contains a router,16 processors, and 16 RAMs, the CM-1 as a whole has a 20-dimensional hypercube-based routing network, a main RAM, and an input-output processor. Each router contains 5 buffers to store the data being transmitted when a clear channel isnt available, the engineers had originally calculated that 7 buffers per chip would be needed, but this made the chip slightly too large too build. Nobel Prize winning physicist Richard Feynman had previously calculated that 5 buffers would be enough and they resubmitted the design of the chip with only 5 buffers, and when they put the machine together, it worked fine. Each chip is connected to a device called a nexus. The CM-1 uses Feynmans algorithm for computing logarithms that he had developed at Los Alamos National Laboratory for the Manhattan Project and it is well suited to the CM-1, using as it did, only shifting and adding, with a small table shared by all the processors. Feynman also discovered that the CM-1 would compute the Feynman diagrams for quantum chromodynamics calculations faster than a special purpose machine developed at Caltech. To improve its commercial viability, TMC launched the CM-2 in 1987, adding Weitek 3132 floating point numeric coprocessors, thirty-two of the original one-bit processors shared each numeric processor. The CM-2 can be configured with up to 512 MB of RAM, two later variants of the CM-2 were also produced, the smaller CM-2a with either 4096 or 8192 single-bit processors, and the faster CM-200. Other early languages included Karl Sims IK and Cliff Lassers URDU, much system utility software for the CM-1/2 was written in *Lisp. Many applications for the CM-2, however, were written in C*, to make programming easier, it was made to simulate a SIMD design. The later CM-5E replaces the SPARC processors with faster SuperSPARCs, the CM-5 was the second-fastest system in the November 1993 TOP500 list, running 1024 cores with Rpeak of 131.0 GFLOPS/s. Connection Machines were noted for their visual design. The CM-1 and CM-2 design teams were led by Tamiko Thiel, by default, when a processor is executing an instruction, its LED is on. In a SIMD program, the goal is to have as many processors as possible working the program at the same time – indicated by having all LEDs being steady on. Those unfamiliar with the use of the LEDs wanted to see the LEDs blink – or even spell out messages to visitors, the result is that finished programs often have superfluous operations to blink the LEDs
5.
Zonohedron
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A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of segments in three-dimensional space. Zonohedra were originally defined and studied by E. S. Fedorov, more generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron, each primary parallelohedron is combinatorially equivalent to one of five types, the rhombohedron, hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron. Let be a collection of three-dimensional vectors, with each vector vi we may associate a line segment. The Minkowski sum forms a zonohedron, and all zonohedra that contain the origin have this form, the vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, by choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. Generators parallel to the edges of an octahedron form a truncated octahedron, the Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Both of these zonohedra are simple, as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles. Any simple zonohedron corresponds in this way to a simplicial arrangement, simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane, which were studied by Grünbaum. There are also many examples that do not fit into these three families. Any prism over a polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular, two faces are equal to the regular polygon from which the prism was formed. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, the truncated cuboctahedron, with 12 squares,8 hexagons, and 6 octagons. The truncated icosidodecahedron, with 30 squares,20 hexagons and 12 decagons, in addition, certain Catalan solids are again zonohedra, The rhombic dodecahedron is the dual of the cuboctahedron. The rhombic triacontahedron is the dual of the icosidodecahedron, zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron
6.
Factorial
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In mathematics, the factorial of a non-negative integer n, denoted by n. is the product of all positive integers less than or equal to n. =5 ×4 ×3 ×2 ×1 =120, the value of 0. is 1, according to the convention for an empty product. The factorial operation is encountered in areas of mathematics, notably in combinatorics, algebra. Its most basic occurrence is the fact there are n. ways to arrange n distinct objects into a sequence. This fact was known at least as early as the 12th century, fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial, Now the nature of these methods is such, the factorial function is formally defined by the product n. = ∏ k =1 n k, or by the relation n. = {1 if n =0. The factorial function can also be defined by using the rule as n. All of the above definitions incorporate the instance 0, =1, in the first case by the convention that the product of no numbers at all is 1. This is convenient because, There is exactly one permutation of zero objects, = n. ×, valid for n >0, extends to n =0. It allows for the expression of many formulae, such as the function, as a power series. It makes many identities in combinatorics valid for all applicable sizes, the number of ways to choose 0 elements from the empty set is =0. More generally, the number of ways to choose n elements among a set of n is = n. n, the factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica, although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n. different ways of arranging n distinct objects into a sequence, often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements, one can obtain such a combination by choosing a k-permutation, successively selecting and removing an element of the set, k times, for a total of n k _ = n ⋯ possibilities. This however produces the k-combinations in an order that one wishes to ignore, since each k-combination is obtained in k. different ways. This number is known as the coefficient, because it is also the coefficient of Xk in n
7.
Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
8.
Cube 2: Hypercube
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It is the second film in the Cube film series, and a sequel to Cube. The film starts with a woman named Becky, shown to be trapped in the Cube. She enters another room, but unknown to her, the room has reversed gravity, as she is pulled up and they come across Colonel Thomas Maguire, who says that they have to solve the code in order to leave the mysterious place. Just as he finishes, a wall begins to close in on the group, the group escapes while Thomas stays behind, while Kate and Simon watch in horror as Thomas is disintegrated by the wall. Kate notices the numbers 60659 everywhere they go, the group soon realizes they are connected to Izon, a defense contractor. While in conversation, Mrs. Paley opens a panel to reveal her being killed by Simon, Jerry thinks that it is a parallel universe, while Max and Julia think it is an optical illusion. Later, while the group is sleeping, Simon realizes that he is in the tesseract to look for Becky, a missing Izon worker, meanwhile, Sasha hears a noise and awakens everyone, due to her acute hearing. The group finds a square in the middle of the room. The group flees into another room, but Jerry is injured by the tesseract, trying to call Jerry, the rest of the group watches as Jerry is shredded by the tesseract, which grows rapidly. Sasha and Kate are separated from the group, as a result, Simon starts to suspect that Mrs. Paley is an undercover spy, so he gags and ties her up, but crystal beams start protruding from the walls. Simon tries to save Mrs. Paley, but when he sees that there isnt enough time, as the clinging Mrs. Paley refuses to release him, Max and Julia, disturbed that Simon killed Mrs. Paley, leave and abandon him. Julia tells Max she must be dreaming, and then kisses him, Simon, alone and hungry, goes insane. He encounters a parallel Jerry and the missing Becky, killing both of them, meanwhile, Kate finds grisly alternate realities of her death in other rooms, and is horrified. Sasha tells Kate that time and space are distorted where they are, Kate, however, still believes that there is a way out. Kate then finds Simon and stabs him in the eye after he grabs her in another room, as Simon appears behind Alex, old and blind in one eye, proving Alexs space-time theory. Alex claims that they are all dead, which causes Simon to snap her neck, Kate finds that the tesseract is shrinking, and kills Simon with the knife. The hypercube starts to wear away, and Kate opens a panel in the bottom, at 6,06,59, she jumps in just when the Hypercube implodes. Kate wakes up in the hands of Izon authorities in an unknown factory and she gives them the necklace, but because of their confidentiality, she is shot in the head and killed by one of the Izon operatives
9.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
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