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
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Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
3.
Alternation (geometry)
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In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for hemi or half, because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a. 2b. 2c is a.3. b.3. c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons, a snub can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces, all truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub and this alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the vertices will not in general create uniform facets. Examples, Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb, an alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell, 4-honeycombs, An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube, cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract. Coxeter also used the operator a, which contains both halves, so retains the original symmetry, for even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron, Norman Johnson extended the use of the altered operator a, b for blended, and c for converted, as, and respectively. The compound polyhedron, stellated octahedron can be represented by a, the star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a, and. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the free edges. A similar operation can truncate alternate vertices, rather than just removing them, below is a set of polyhedra that can be generated from the Catalan solids. These have two types of vertices which can be alternately truncated, truncating the higher order vertices and both vertex types produce these forms, Conway polyhedral notation Wythoff construction Coxeter, H. S. M
4.
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
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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
6.
Tetrahedral-octahedral honeycomb
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The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1,2, other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual dodecahedrille and it is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is also part of another family of uniform honeycombs called simplectic honeycombs. In this case of 3-space, the honeycomb is alternated, reducing the cubic cells to tetrahedra. As such it can be represented by an extended Schläfli symbol h as containing half the vertices of the cubic honeycomb, theres a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges, a second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells, the alternated cubic honeycomb can be orthogonally projected into the planar square tiling by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the cubic honeycomb creates two offset copies of the square tiling vertex arrangement of the plane, Its vertex arrangement represents an A3 lattice or D3 lattice. It is the 3-dimensional case of a simplectic honeycomb and its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb. The D+3 packing can be constructed by the union of two D3 lattices, the D+ n packing is only a lattice for even dimensions. The kissing number of the D*3 lattice is 8 and its Voronoi tessellation is a cubic honeycomb. The, Coxeter group generates 15 permutations of uniform honeycombs,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform honeycombs,4 with distinct geometry including the alternated cubic honeycomb
7.
Cubic crystal system
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In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals, there are three main varieties of these crystals, Primitive cubic Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
8.
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
9.
Honeycomb (geometry)
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In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space, honeycombs are usually constructed in ordinary Euclidean space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. There are infinitely many honeycombs, which have only been partially classified, the more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane, in particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary space. Another interesting family is the Hill tetrahedra and their generalizations, which can tile the space. A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of polyhedral cells. There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs, a honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform, however, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. An infinite number of unique honeycombs can be created by order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric, in the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra, Cubic honeycomb Hexagonal prismatic honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb, bitruncated cubic honeycomb Other known examples of space-filling polyhedra include, The Triangular prismatic honeycomb. The gyrated triangular prismatic honeycomb The triakis truncated tetrahedral honeycomb, the Voronoi cells of the carbon atoms in diamond are this shape. The trapezo-rhombic dodecahedral honeycomb Isohedral tilings, sometimes, two or more different polyhedra may be combined to fill space. Two classes can be distinguished, Non-convex cells which pack without overlapping and these include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities cancel out to form a uniformly dense continuum, in 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size
10.
Sphere packing
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In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space, however, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the fill as large a proportion of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement, for equal spheres in three dimensions the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%, a lattice arrangement is one in which the centers of the spheres form a very symmetric pattern which only needs n vectors to be uniquely defined. Arrangements in which the spheres do not form a lattice can still be periodic, lattice arrangements are easier to handle than irregular ones—their high degree of symmetry makes it easier to classify them and to measure their densities. In three-dimensional Euclidean space, the densest packing of spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows, consider a plane with a compact arrangement of spheres on it. For any three neighbouring spheres, a sphere can be placed on top in the hollow between the three bottom spheres. If we do this everywhere in a plane above the first. A third layer can be placed directly above the first one, or the spheres can be offset, there are thus three types of planes, called A, B and C. Two simple arrangements within the close-packed family correspond to regular lattices, one is called cubic close packing — where the layers are alternated in the ABCABC… sequence. The other is called hexagonal close packing — where the layers are alternated in the ABAB… sequence, but many layer stacking sequences are possible, and still generate a close-packed structure. In all of these arrangements each sphere is surrounded by 12 other spheres, carl Friedrich Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings. In 1611 Johannes Kepler had conjectured that this is the maximum possible density amongst both regular and irregular arrangements — this became known as the Kepler conjecture. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, Hales proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were 99% certain of the correctness of Hales proof, on 10 August 2014 Hales announced the completion of a formal proof using automated proof checking, removing any doubt. Some other lattice packings are often found in physical systems, Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or jammed
11.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
12.
Voronoi diagram
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In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points is specified beforehand, and for each seed there is a region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells, the Voronoi diagram of a set of points is dual to its Delaunay triangulation. It is named after Georgy Voronoi, and is called a Voronoi tessellation, a Voronoi decomposition. Voronoi diagrams have practical and theoretical applications to a number of fields, mainly in science and technology. They are also known as Thiessen polygons, in the simplest case, shown in the first picture, we are given a finite set of points in the Euclidean plane. Each such cell is obtained from the intersection of half-spaces, the line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices are the points equidistant to three sites, let X be a metric space with distance function d. Let K be a set of indices and let k ∈ K be a tuple of nonempty subsets in the space X. In other words, if d = inf denotes the distance between the point x and the subset A, then R k = The Voronoi diagram is simply the tuple of cells k ∈ K. In principle some of the sites can intersect and even coincide, in addition, infinitely many sites are allowed in the definition, but again, in many cases only finitely many sites are considered. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram, however, in general the Voronoi cells may not be convex or even connected. In the usual Euclidean space, we can rewrite the definition in usual terms. Each Voronoi polygon R k is associated with a generator point P k, let X be the set of all points in the Euclidean space. Let P1 be a point that generates its Voronoi region R1, P2 that generates R2, and P3 that generates R3, and so on. Then, as expressed by Tran et al all locations in the Voronoi polygon are closer to the point of that polygon than any other generator point in the Voronoi diagram in Euclidian plane. As a simple illustration, consider a group of shops in a city, suppose we want to estimate the number of customers of a given shop. With all else being equal, it is reasonable to assume that customers choose their preferred shop simply by distance considerations, they will go to the shop located nearest to them