Some special cases are: angle of 0° or 180° (degenerate case) two non-adjacent sides are on the same line equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32) equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32) A triangle is equilateral iff it is equiangular. An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "ang ...

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Polygon, Polygon - Names and types, Polygon - Naming polygons, Polygon - Taxonomic classification, Polygon - Properties, Polygon - Angles, Polygon - Area, Polygon - Construction, Polygon - Point in polygon test, Polygon - Special cases

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A monotone polygon is one with a boundary that consists of two parts, each of which consists of points that have incrementing coordinates in one dimension. Such a polygon can easily be triangulated in linear time as described by A. Fournier and D.Y. Montuno. To break up a polygon into monotone polygons, follow these steps: For each point, check if the vertices are both on the same side of the 'sweep line', a horizontal or vertical line. If they are, check the next sweep line on the other side. Break the polygon on the l ...

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Polygon triangulation, Polygon triangulation - Substracting ears method, Polygon triangulation - Using monotone polygons, Polygon triangulation - Reference

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Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula. Polygon - Naming polygons. To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows That is, a 42-sided figure would be nam ...

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Polygon, Polygon - Names and types, Polygon - Naming polygons, Polygon - Taxonomic classification, Polygon - Properties, Polygon - Angles, Polygon - Area, Polygon - Construction, Polygon - Point in polygon test, Polygon - Special cases

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One way to triangulate a simple polygon is by using the assertion that any simple polygon without holes has at least two so called 'ears'. An ear is a triangle with two sides on the edge of the polygon and the other one completely inside it. The algorithm then consists of finding such an ear, removing it from the polygon (which results in a new polygon that still meets the conditions) and repeating until there is only one triangle left. This algorithm is pretty easy to implement, but imposes restrictions o ...

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Polygon triangulation, Polygon triangulation - Substracting ears method, Polygon triangulation - Using monotone polygons, Polygon triangulation - Reference

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Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the c ...

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Constructible polygon, Constructible polygon - Conditions for constructibility, Constructible polygon - General theory, Constructible polygon - Detailed results in terms of Fermat primes, Constructible polygon - Compass-and-straightedge constructions, Constructible polygon - Other constructions

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In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence of quadratic equations. In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have degree over the base field that is a power of two. In the specific case of a regular n-gon, the question reduces to the question of cons ...

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Constructible polygon, Constructible polygon - Conditions for constructibility, Constructible polygon - General theory, Constructible polygon - Detailed results in terms of Fermat primes, Constructible polygon - Compass-and-straightedge constructions, Constructible polygon - Other constructions

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One simple way of finding whether the point is inside or outside a simple polygon is to test how many times a ray starting from the point intersects the edges of the polygon. If it is an even number of times, the point is outside, if odd, inside. Below is another variant of this approach. Let P = (x0,y0) be the point to be tested against a polygon specified by the ordered list of its vertices (and hence, edges). Set l := 0 and r := 0 for ea ...

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Point in polygon, Point in polygon - Ray casting algorithms, Point in polygon - External link

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Compass-and-straightedge constructions are known for all constructible polygons. If n = p·q with p = 2 or p and q coprime, an n-gon can be constructed from a p-gon and a q-gon. If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be constructed. If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a vertex. ...

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Constructible polygon, Constructible polygon - Conditions for constructibility, Constructible polygon - General theory, Constructible polygon - Detailed results in terms of Fermat primes, Constructible polygon - Compass-and-straightedge constructions, Constructible polygon - Other constructions

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The internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular n-gon has internal angle degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex. Only fifteen of these can occur in any tiling of regular polygons. For example, 3.7.42 cannot occur because it is the onl ...

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Tiling by regular polygons, Tiling by regular polygons - Regular tilings, Tiling by regular polygons - Archimedean uniform or semiregular tilings, Tiling by regular polygons - Combinations of regular polygons that can meet at a vertex, Tiling by regular polygons - Other edge-to-edge tilings, Tiling by regular polygons - Tilings that are not edge-to-edge, Tiling by regular polygons - Beyond the plane

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Only five Fermat primes are known: F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in OEIS). Thus an n-gon is constructible if n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ... (sequence A003401 in OEIS), while and an n-gon is not constructible with compass and straightedge if n = 7, 9 ...

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Constructible polygon, Constructible polygon - Conditions for constructibility, Constructible polygon - General theory, Constructible polygon - Detailed results in terms of Fermat primes, Constructible polygon - Compass-and-straightedge constructions, Constructible polygon - Other constructions

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These tessellations are also related to regular and semiregular polyhedra and tessellations of the hyperbolic plane. Semiregular polyhedra are made from regular polygon faces, but their angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry have angles smaller than they do in the plane. In both these cases, that the arrangement of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive. Some regular tilings ...

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Tiling by regular polygons, Tiling by regular polygons - Regular tilings, Tiling by regular polygons - Archimedean uniform or semiregular tilings, Tiling by regular polygons - Combinations of regular polygons that can meet at a vertex, Tiling by regular polygons - Other edge-to-edge tilings, Tiling by regular polygons - Tilings that are not edge-to-edge, Tiling by regular polygons - Beyond the plane

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Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by regular polygons may be drawn. Here are four examples: Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are n orbits of vertices, a tiling is known as n-uniform or n-isogonal; if there are n orbits of tiles, as n-isohedral; i ...

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Tiling by regular polygons, Tiling by regular polygons - Regular tilings, Tiling by regular polygons - Archimedean uniform or semiregular tilings, Tiling by regular polygons - Combinations of regular polygons that can meet at a vertex, Tiling by regular polygons - Other edge-to-edge tilings, Tiling by regular polygons - Tilings that are not edge-to-edge, Tiling by regular polygons - Beyond the plane

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Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. The above models can be described as follows: sphere: AA − ...

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Surface, Surface - Examples, Surface - Definition, Surface - Classification of closed surfaces, Surface - Compact surfaces, Surface - Embeddings in R3, Surface - Differential geometry, Surface - Some models, Surface - Fundamental polygon, Surface - Connected sum of surfaces, Surface - Algebraic surface

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Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second. If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as Archimedean, uniform or semiregular tilings. Grünbaum and Shephard distinguish the description of these tilings as Archimedean as referring only to the local property of the arran ...

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Tiling by regular polygons, Tiling by regular polygons - Regular tilings, Tiling by regular polygons - Archimedean uniform or semiregular tilings, Tiling by regular polygons - Combinations of regular polygons that can meet at a vertex, Tiling by regular polygons - Other edge-to-edge tilings, Tiling by regular polygons - Tilings that are not edge-to-edge, Tiling by regular polygons - Beyond the plane

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The Schläfli symbol of a polygon with n edges is {n}. For example {5} is a pentagon. There are nonconvex polygons which are considered regular. They are called star polygons. A star polygon with symbol {p/q} has p vertices where every q-th vertex is connected. Thus, 5/2 is a pentagram. ...

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Schläfli symbol, Schläfli symbol - Regular polygons plane, Schläfli symbol - Regular polyhedra 3-space, Schläfli symbol - Regular polychora 4-space, Schläfli symbol - Higher dimensions, Schläfli symbol - Dual polytopes

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If all faces are regular polygons, the pyramid base can be a regular polygon of 3, 4 or 5 sided: The geometric center of a square-based pyramid is located on the symmetry axis, one quarter of the way from the base to the apex. Pyramid geometry - Symmetry. If the base is regular and the apex is above the center, the symmetry group of the n-sided pyramid is Cnv of order 2n, except in the case of a regular tetrahedron, which has the larger symmetry group Td ...

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Pyramid geometry, Pyramid geometry - Volume, Pyramid geometry - Pyramids with regular polygon faces, Pyramid geometry - Symmetry

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An n-sided regular polygon can be constructed with ruler and compass if and only if n is a power of 2 or the product of a power of 2 and distinct Fermat primes. In other words, if and only if n is of the form n = 2kp1p2...ps, where k is a nonnegative integer and the pi are distinct Fermat primes. See constructible polygon. A positive integer n is of the above form if and only if φ(See also:

Fermat number, Fermat number - Basic properties, Fermat number - Primality of Fermat numbers, Fermat number - Factorisation of Fermat numbers, Fermat number - Fermat's little theorem and pseudoprimes, Fermat number - Other theorems about Fermat's primes, Fermat number - Relationship to constructible polygons, Fermat number - Applications of Fermat numbers, Fermat number - Other interesting facts, Fermat number - Generalised Fermat numbers

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A polygonal fort is a fortification in the style that evolved around the middle of the nineteenth century, in response to the development of powerfull explosive shells. The complex and sophisticated designs of star forts that preceded them were highly effective against cannon assault, but proved much less effective against the more accurate fire of rifled guns and the destructive power of explosive shells. The polygonal style of fortification is also described as a "flankless fort". Many were built during the government of Lord ...

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Gunpowder warfare, Gunpowder warfare - Cannons, Gunpowder warfare - Beginning of polygonal fortifications, Gunpowder warfare - Firearms, Gunpowder warfare - Nature of war, Gunpowder warfare - Outside of Europe, Gunpowder warfare - Ottoman Empire, Gunpowder warfare - Japan, Gunpowder warfare - Naval warfare

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Some regular polygons (e.g. a pentagon) are easy to construct with ruler and compass; others are not. This led to the question: Is it possible to construct all regular polygons with ruler and compass? Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary, but he offered no proof of thi ...

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Ruler-and-compass construction, Ruler-and-compass construction - Ruler and compass, Ruler-and-compass construction - Constructible points and lengths, Ruler-and-compass construction - Impossible constructions, Ruler-and-compass construction - Squaring the circle, Ruler-and-compass construction - Doubling the cube, Ruler-and-compass construction - Angle trisection, Ruler-and-compass construction - Constructing regular polygons, Ruler-and-compass construction - Constructing with only ruler or only compass, Ruler-and-compass construction - Recent research, Ruler-and-compass construction - Reference

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For an infinite tiling, let a be the average number of sides of a polygon, and b the average number of sides meeting at a vertex. Then ( a − 2 ) ( b − 2 ) = 4. For example, we have the combinations (3,6), (3 1/3 , 5), (3 3/4, 4 2/7), (4,4), and (6,3) for the tilings in the article Tilings of regular polygons. A continuation of a side in a straight line beyond a vertex is counted as a separate side. For example, the bricks in the picture are considered hexagons, and ...

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Tessellation, Tessellation - Wallpaper groups, Tessellation - Tessellations and color, Tessellation - Tessellations with quadrilaterals, Tessellation - Regular and irregular tessellations, Tessellation - Tessellations and computer graphics, Tessellation - Number of sides of a polygon versus number of sides at a vertex, Tessellation - History

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By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs (x,x), is called the diagonal. It is the graph of the identity relation. It plays an important part in geometry: for example the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. Quite a major role is played in geometric studies by the idea of intersecting the diagonal with itself: not directly, but by perturbing it wi ...

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Diagonal, Diagonal - Polygons, Diagonal - Matrices, Diagonal - Geometry

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