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17th Century In Chess
17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number. 17 was described at MIT as "the least random number", according to the Jargon File. This is supposedly because, in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice. This study has been repeated a number of times. Mathematics 17 is a Leyland number and Leyland prime, using 2 & 3 (23 + 32) and using 4 and 5, using 3 & 4 (34 - 43). 17 is a Fermat prime. 17 is one of six lucky numbers of Euler. Since seventeen is a Fermat prime, regular heptadecagons can be constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies. The minimum possible number of givens for a sudoku puzzle with a unique solution is 17. Geometric properties Two-dimensions *There are seventeen crystallographic space groups in two dimensions. These are som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheel Of Theodorus
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene. Construction The spiral is started with an isosceles right triangle, with each Cathetus, leg having unit length. Another right triangle (which is the ''only'' automedian triangle, automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3. The process then repeats; the nth triangle in the sequence is a right triangle with the side lengths \sqrt and 1, and with hypotenuse \sqrt. For example, the 16th triangle has sides measuring 4=\sqrt, 1 and hypotenuse of \sqrt. History and uses Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue ''Theae ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R(3, 3, 3) = 17
R, or r, is the eighteenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ar'' (pronounced ), plural ''ars''. The letter is the eighth most common letter in English and the fourth-most common consonant, after , , and . Name The name of the letter in Latin was (), following the pattern of other letters representing continuants, such as , , , , and . This name is preserved in French and many other languages. In Middle English, the name of the letter changed from to , following a pattern exhibited in many other words such as ''farm'' (compare French ) and ''star'' (compare German ). In Hiberno-English, the letter is called or , somewhat similar to ''oar'', ''ore'', ''orr''. The letter R is sometimes referred to as the 'canine letter', often rendered in English as the dog's letter. This Latin term referred to the Latin that was trilled to sound l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monochromatic Triangle
In graph theory and theoretical computer science, the monochromatic triangle problem is an algorithmic problem on graphs, in which the goal is to partition the edges of a given graph into two triangle-free subgraphs. It is NP-complete but fixed-parameter tractable on graphs of bounded treewidth. Problem statement The monochromatic triangle problem takes as input an n-node undirected graph G(V,E) with node set V and edge set E. The output is a Boolean value, true if the edge set E of G can be partitioned into two disjoint sets E1 and E2, such that both of the two subgraphs G1(V,E1) and G2(V,E2) are triangle-free graphs, and false otherwise. This decision problem is NP-complete. Generalization to multiple colors The problem may be generalized to triangle-free edge coloring, finding an assignment of colors to the edges of a graph so that no triangle has all three edges given the same color. The monochromatic triangle problem is the special case of triangle-free edge-coloring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertex (geometry), vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two Face (geometry), faces (or polyhedron sides) meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. An ''edge'' may also be an infinite line (geometry), line separating two half-planes. The ''sides'' of a plane angle are semi-infinite Half-line (geometry), half-lines (or rays). Relation to edges in graphs In graph theory, an Edge (graph theory), edge is an abstract object connecting two vertex (graph theory), graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its n-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosagon
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees. Regular icosagon The Regular polygon, regular icosagon has Schläfli symbol , and can also be constructed as a Truncation (geometry), truncated decagon, , or a twice-truncated pentagon, . One interior angle in a regular polygon, regular icosagon is 162°, meaning that one exterior angle would be 18°. The area of a regular icosagon with edge length is :A=t^2(1+\sqrt+\sqrt) \simeq 31.5687 t^2. In terms of the radius of its circumcircle, the area is :A=\frac(\sqrt-1); since the area of the circle is \pi R^2, the regular icosagon fills approximately 98.36% of its circumcircle. Uses The Big Wheel on the popular US game show ''The Price Is Right (U.S. game show), The Price Is Right'' has an icosagonal cross-section. The Globe, the outdoor theater used by William Shakespeare's acting company, was discovered to have been built on an icosagonal foundation when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentadecagon
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Regular pentadecagon A '' regular pentadecagon'' is represented by Schläfli symbol . A regular pentadecagon has interior angles of 156 °, and with a side length ''a'', has an area given by : \begin A = \fraca^2 \cot \frac & = \frac\sqrta^2 \\ & = \frac \left( \sqrt+\sqrt+ \sqrt\sqrt \right) \\ & \simeq 17.6424\,a^2. \end Construction As 15 = 3 × 5, a product of distinct Fermat primes, a regular pentadecagon is constructible using compass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's ''Elements''. Compare the construction according to Euclid in this imagePentadecagon In the construction for given circumcircle: \overline = \overline\text \; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octadecagon
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. Regular octadecagon A ''regular polygon, regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular Truncation (geometry), truncated enneagon, t, which alternates two types of edges. Construction As 18 = 2 × 32, a regular octadecagon cannot be constructible polygon, constructed using a compass and straightedge. However, it is constructible using neusis construction, neusis, or an angle trisection with a Tomahawk_(geometry), tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. Symmetry The ''regular octadecagon'' has dihedral symmetry, Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9), (Z6, Z3), an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icositetragon
In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees. Regular icositetragon The ''regular polygon, regular icositetragon'' is represented by Schläfli symbol and can also be constructed as a Truncation (geometry), truncated dodecagon, t, or a twice-truncated hexagon, tt, or thrice-truncated triangle, ttt. One interior angle in a regular polygon, regular icositetragon is 165°, meaning that one exterior angle would be 15°. The area of a regular icositetragon is: (with ''t'' = edge length) : A = 6t^2 \cot \frac = t^2(2+\sqrt+\sqrt+\sqrt). The icositetragon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), tetracontaoctagon (48-gon), and enneacontahexagon (96-gon). Construction As 24 = 23 × 3, a regular icositetragon is constructible polygon, constructible using a compass and straightedge. As a truncated dodecagon, it ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Tilings Of Convex Regular Polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his (Latin: ''The Harmony of the World'', 1619). Notation of Euclidean tilings Euclidean tilings are usually named after Cundy & Rollett’s notation. This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 36; 36; 34.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 36; 36 (both of different transitivity class), or (36)2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 34.6, 4 more contiguous equilateral triangles and a single regular hexagon. However, this notation has two main problems rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (: vertices or vertexes), also called a corner, is a point (geometry), point where two or more curves, line (geometry), lines, or line segments Tangency, meet or Intersection (geometry), intersect. For example, the point where two lines meet to form an angle and the point where edge (geometry), edges of polygons and polyhedron, polyhedra meet are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
