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30-398: 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

60-597: A mirror plane of symmetry S 1 , an inversion center of symmetry S 2 , or a higher improper rotation (rotoreflection) S n axis of symmetry is achiral. (A plane of symmetry of a figure F {\displaystyle F} is a plane P {\displaystyle P} , such that F {\displaystyle F} is invariant under the mapping ( x , y , z ) ↦ ( x , y , − z ) {\displaystyle (x,y,z)\mapsto (x,y,-z)} , when P {\displaystyle P}

90-554: 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

120-402: 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

150-424: 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

180-410: 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 ,

210-457: A right or left handedness , according to the right-hand rule . Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out . The J-, L-, S- and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in

240-424: A two-dimensional space. Individually they contain no mirror symmetry in the plane. A figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. (In Euclidean geometry any isometry can be written as v ↦ A v + b {\displaystyle v\mapsto Av+b} with an orthogonal matrix A {\displaystyle A} and

270-404: A vector b {\displaystyle b} . The determinant of A {\displaystyle A} is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving. A general definition of chirality based on group theory exists. It does not refer to any orientation concept: an isometry is direct if and only if it

300-522: 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

330-492: Is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime. In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. (An axis of symmetry of a figure F {\displaystyle F} is a line L {\displaystyle L} , such that F {\displaystyle F}

SECTION 10

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360-506: 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

390-554: 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: Chirality (mathematics) In geometry , a figure is chiral (and said to have chirality ) if it is not identical to its mirror image , or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that

420-404: Is chiral if it is scalene . Consider the following pattern: This figure is chiral, as it is not identical to its mirror image: But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a frieze group generated by a single glide reflection . In three dimensions, every figure that possesses

450-600: Is chosen to be the x {\displaystyle x} - y {\displaystyle y} -plane of the coordinate system. A center of symmetry of a figure F {\displaystyle F} is a point C {\displaystyle C} , such that F {\displaystyle F} is invariant under the mapping ( x , y , z ) ↦ ( − x , − y , − z ) {\displaystyle (x,y,z)\mapsto (-x,-y,-z)} , when C {\displaystyle C}

480-487: Is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure which is invariant under the orientation reversing isometry ( x , y , z ) ↦ ( − y , x , − z ) {\displaystyle (x,y,z)\mapsto (-y,x,-z)} and thus achiral, but it has neither plane nor center of symmetry. The figure also

510-399: Is invariant under the mapping ( x , y ) ↦ ( x , − y ) {\displaystyle (x,y)\mapsto (x,-y)} , when L {\displaystyle L} is chosen to be the x {\displaystyle x} -axis of the coordinate system.) For that reason, a triangle is achiral if it is equilateral or isosceles , and

540-449: Is not chiral is said to be achiral . A chiral object and its mirror image are said to be enantiomorphs . The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'. Some chiral three-dimensional objects, such as the helix , can be assigned

570-475: 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

600-510: 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

630-428: 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

SECTION 20

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660-413: 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

690-415: 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

720-440: 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

750-530: 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

780-499: 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

810-504: 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

840-475: 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

870-401: 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 ,

900-433: 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

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