Misplaced Pages

#227772

33-398: A group of twenty units may be referred to as a score . Twenty is a pronic number , as it is the product of consecutive integers, namely 4 and 5. It is also the second pronic sum number (or pronic pyramid) after 2, being the sum of the first three pronic numbers: 2 + 6 + 12. It is the third composite number to be the product of a squared prime and a prime (and also the second member of

66-565: A convention that is adopted in the following sections. The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle 's Metaphysics , and their discovery has been attributed much earlier to the Pythagoreans . As a kind of figurate number, the pronic numbers are sometimes called oblong because they are analogous to polygonal numbers in this way: The n th pronic number

99-402: A mirror. However, not all of them can be constructed in such a way, or they could be constructed alternatively. For example, the icosidodecahedron can be constructed by attaching two pentagonal rotunda base-to-base, or rhombicuboctahedron that can be constructed alternatively by attaching two square cupolas on the bases of octagonal prism. There are at least for known ten solids that have

132-632: A now-lost work. Although they were not credited to Archimedes originally, Pappus of Alexandria in the fifth section of his titled compendium Synagoge referring that Archimedes listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have. During the Renaissance , artists and mathematicians valued pure forms with high symmetry. Some Archimedean solids appeared in Piero della Francesca 's De quinque corporibus regularibus , in attempting to study and copy

165-402: A perfect square, and the n th perfect square is at a radius of n from a pronic number. The n th pronic number is also the difference between the odd square (2 n + 1) and the ( n +1) st centered hexagonal number . Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number. The partial sum of the first n positive pronic numbers

198-424: A pronic number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a pronic number is present in only one of the factors n or n + 1 . Thus a pronic number is squarefree if and only if n and n + 1 are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n + 1 . If 25

231-410: A regular compound of five octahedra . In total, there are 20 semiregular polytopes that only exist up through the 8th dimension, which include 13 Archimedean solids and 7 Gosset polytopes (without counting enantiomorphs , or semiregular prisms and antiprisms). The Happy Family of sporadic groups is made up of twenty finite simple groups that are all subquotients of the friendly giant ,

264-522: Is a number that is the product of two consecutive integers , that is, a number of the form n ( n + 1 ) {\displaystyle n(n+1)} . The study of these numbers dates back to Aristotle . They are also called oblong numbers , heteromecic numbers , or rectangular numbers ; however, the term "rectangular number" has also been applied to the composite numbers . The first few pronic numbers are: Letting P n {\displaystyle P_{n}} denote

297-511: Is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number . The arithmetic mean of two consecutive pronic numbers is a square number : So there is a square between any two consecutive pronic numbers. It is unique, since Another consequence of this chain of inequalities is the following property. If m is a pronic number, then the following holds: The fact that consecutive integers are coprime and that

330-587: Is appended to the decimal representation of any pronic number, the result is a square number, the square of a number ending on 5; for example, 625 = 25 and 1225 = 35 . This is so because The difference between two consecutive unit fractions is the reciprocal of a pronic number : Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes , although he did not claim credit for them. They belong to

363-491: Is not vertex-transitive . The Archimedean solids have the vertex configuration and highly symmetric properties. Vertex configuration means a polyhedron whose two or more polygonal faces meet at the vertex. For instance, the 3 ⋅ 5 ⋅ 3 ⋅ 5 {\displaystyle 3\cdot 5\cdot 3\cdot 5} means a polyhedron in which each vertex is met by alternating two triangles and two pentagons. Highly symmetric properties in this case mean

SECTION 10

#1732764810228

396-479: Is the length of a side of the fifth smallest right triangle that forms a primitive Pythagorean triple (20, 21 , 29 ). It is the third tetrahedral number . In combinatorics , 20 is the number of distinct combinations of 6 items taken 3 at a time. Equivalently, it is the central binomial coefficient for n=3 (sequence A000984 in the OEIS ). In decimal , 20 is the smallest non-trivial neon number equal to

429-519: Is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. 20 is the third magic number in physics. In chemistry , it is the atomic number of calcium . Formerly the age of majority in Japan and in Japanese tradition. 20 is the basis for vigesimal number systems, used by several different civilizations in the past (and to this day), including

462-428: Is the sum of the first n even integers, and as such is twice the n th triangular number and n more than the n th square number , as given by the alternative formula n + n for pronic numbers. Hence the n th pronic number and the n th square number (the sum of the first n odd integers ) form a superparticular ratio : Due to this ratio, the n th pronic number is at a radius of n and n + 1 from

495-413: Is twenty faces, which make up a regular icosahedron . A dodecahedron , on the other hand, has twenty vertices, likewise the most a regular polyhedron can have. There are a total of 20 regular and semiregular polyhedra, aside from the infinite family of semiregular prisms and antiprisms that exists in the third dimension: the 5 Platonic solids, and 15 Archimedean solids (including chiral forms of

528-415: Is twice the value of the n th tetrahedral number : The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1: The partial sum of the first n terms in this series is The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series : Pronic numbers are even, and 2 is the only prime pronic number. It

561-440: The 2 × q family in this form). It is a largely composite number , as it has 6 divisors and no smaller number has more than 6 divisors. It has an aliquot sum of 22 ; a semiprime , within an aliquot sequence of four composite numbers (20, 22, 14 , 10 , 8 ) that belong to the prime 7 -aliquot tree. It is the smallest primitive abundant number , and the first number to have an abundance of 2 , followed by 104 . 20

594-534: The Maya . Les XX ("The 20") was a group of twenty Belgian painters, designers and sculptors, formed in 1883. In chess , 20 is the number of legal moves for each player in the starting position. A 'score' is a group of twenty (often used in combination with a cardinal number , e.g. fourscore to mean 80), but also often used as an indefinite number (e.g. the newspaper headline "Scores of Typhoon Survivors Flown to Manila"). Pronic number A pronic number

627-552: The Rupert property , a polyhedron that can pass through a copy of itself with the same or similar size. They are the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron, and the truncated tetrahedron. The Catalan solids are the dual polyhedron of Archimedean solids. The names of Archimedean solids were taken from Ancient Greek mathematician Archimedes , who discussed them in

660-499: The snub cube and snub dodecahedron ). There are also four uniform compound polyhedra that contain twenty polyhedra ( UC 13 , UC 14 , UC 19 , UC 33 ), which is the most any such solids can have; while another twenty uniform compounds contain five polyhedra (that are not part of classes of infinite families, where there exist three more). The compound of twenty octahedra can be obtained by orienting two pairs of compounds of ten octahedra , which can also coincide to yield

693-528: The symmetry group of each solid were derived from the Platonic solids , resulting from their construction. Some sources say the Archimedean solids are synonymous with the semiregular polyhedron . Yet, the definition of a semiregular polyhedron may also include the infinite prisms and antiprisms , including the elongated square gyrobicupola . The construction of some Archimedean solids begins from

SECTION 20

#1732764810228

726-535: The Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra, as well as defining the prisms , antiprisms , and the non-convex solids known as Kepler–Poinsot polyhedra . Kepler may have also found another solid known as elongated square gyrobicupola or pseudorhombicuboctahedron . Kepler once stated that there were fourteen Archimedean solids, yet his published enumeration only includes

759-469: The Platonic solids. The truncation involves cutting away corners; to preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners, and an example can be found in truncated icosahedron constructed by cutting off all the icosahedron 's vertices, having the same symmetry as the icosahedron, the icosahedral symmetry . If

792-402: The centroid and filling them with squares. Snub is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with equilateral triangles . Examples can be found in snub cube and snub dodecahedron . The resulting construction of these solids gives the property of chiral , meaning they are not identical when reflected in

825-424: The class of uniform polyhedra , the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance . The elongated square gyrobicupola or pseudo­rhombi­cub­octa­hedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it

858-504: The largest of twenty-six sporadic groups. The largest supersingular prime factor that divides the order of the friendly giant is 71 , which is the 20th indexed prime number, where 26 also represents the number of partitions of 20 into prime parts. Both 71 and 20 represent self-convolved Fibonacci numbers, respectively the seventh and fifth members j {\displaystyle j} in this sequence F j 2 {\displaystyle F_{j}^{2}} . 20

891-502: The other one, but the elongated square gyrobicupola does not. Grünbaum (2009) observed that it meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit

924-481: The plane containing 2 orbits of vertices . 20 is the number of parallelogram polyominoes with 5 cells. Bring's curve is a Riemann surface of genus four, whose fundamental polygon is a regular hyperbolic twenty-sided icosagon , with an area equal to 12 π {\displaystyle 12\pi } by the Gauss-Bonnet theorem . The largest number of faces a Platonic solid can have

957-401: The pronic number n ( n + 1 ) {\displaystyle n(n+1)} , we have P − n = P n − 1 {\displaystyle P_{{-}n}=P_{n{-}1}} . Therefore, in discussing pronic numbers, we may assume that n ≥ 0 {\displaystyle n\geq 0} without loss of generality ,

990-433: The sum of its digits when raised to the thirteenth power (20 = 8192 × 10). Gelfond's constant and pi very nearly have a difference equal to twenty: differing only by about − 0.000900020811 … {\displaystyle -0.000900020811\ldots } from an integer value. There are twenty edge-to-edge 2-uniform tilings by convex regular polygons, which are uniform tessellations of

1023-442: The thirteen uniform polyhedra. The first clear statement of such solid existence was made by Duncan Sommerville in 1905. The solid appeared when some mathematicians mistakenly constructed the rhombicuboctahedron : two square cupolas attached to the octagonal prism , with one of them rotated in forty-five degrees. The thirteen solids have the property of vertex-transitive , meaning any two vertices of those can be translated onto

20 (number) - Misplaced Pages Continue

1056-407: The truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification . Expansion involves moving each face away from the center (by the same distance to preserve the symmetry of the Platonic solid) and taking the convex hull. An example is the rhombicuboctahedron, constructed by separating the cube or octahedron's faces from

1089-594: The works of Archimedes, as well as include citations to Archimedes. Yet, he did not credit those shapes to Archimedes and know of Archimedes' work but rather appeared to be an independent rediscovery. Other appearance of the solids appeared in the works of Wenzel Jamnitzer 's Perspectiva Corporum Regularium , and both Summa de arithmetica and Divina proportione by Luca Pacioli , drawn by Leonardo da Vinci . The net of Archimedean solids appeared in Albrecht Dürer 's Underweysung der Messung , copied from

#227772