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Centered Polyhedral Number
In mathematics, the centered polyhedral numbers are a class of figurate numbers, each formed by a central dot, surrounded by polyhedral layers with a constant number of edges. The length of the edges increases by one in each additional layer. Examples * Centered tetrahedral numbers * Centered cube numbers * Centered octahedral number In mathematics, a centered octahedral number or Haüy octahedral number is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases ...s * Centered dodecahedral numbers * Centered icosahedral numbers * Stella octangula numbers References * {{Num-stub Figurate numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figurate Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathematicians already considered triangular numbers, polygonal numbers, tetrahedral numbers, and pyramidal numbers, ReprintedG. E. Stechert & Co., 1934 and AMS Chelsea Publishing, 1944. and subsequent mathematicians have included other classes of these numbers including numbers defined from other types of polyhedra and from their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Tetrahedral Number
In mathematics, a centered tetrahedral number is a centered figurate number that represents a tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet .... That is, it counts the dots in a three-dimensional dot pattern with a single dot surrounded by tetrahedral shells. The nth centered tetrahedral number, starting at n=0 for a single dot, is:Deza numbers the centered tetrahedral numbers at n=1 for a single dot, leading to a different formula. The first such numbers are: References Figurate numbers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Cube Number
A centered cube number is a centered figurate number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with points on the square faces of the th layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has points along each of its edges. The first few centered cube numbers are : 1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... . Formulas The centered cube number for a pattern with concentric layers around the central point is given by the formula :n^3 + (n + 1)^3 = (2n+1)\left(n^2+n+1\right). The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as :\binom-\binom = (n^2+1)+(n^2+2)+\cdots+(n+1)^2. Properties Because of the factorization , it is impossible for a centered cube number to be a prime number. The only centered cube numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Octahedral Number
In mathematics, a centered octahedral number or Haüy octahedral number is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. The Haüy octahedral numbers are named after René Just Haüy. History The name "Haüy octahedral number" comes from the work of René Just Haüy, a French mineralogist active in the late 18th and early 19th centuries. His "Haüy construction" approximates an octahedron as a polycube, formed by accreting concentric layers of cubes onto a central cube. The centered octahedral numbers count the cubes used by this construction. Haüy proposed this construction, and several related constructions of other polyhedra, as a model for the structure of crystalline minerals.. See in particulapp. 13–14 As cited by Formula The number of three-dimensional lattice point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Dodecahedral Number
In mathematics, a centered dodecahedral number is a centered figurate number that represents a dodecahedron. The centered dodecahedral number for a specific ''n'' is given by :(2n+1)\left(5n^2+5n+1\right) The first such numbers are: 1, 33, 155 Year 155 ( CLV) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Rufinus (or, less frequently, year 908 ''Ab urbe condita''). The denomination 155 for this year ..., 427, 909, 1661, 2743, 4215, 6137, 8569, … . Congruence Relations * CDC(n) \equiv 1 \pmod * CDC(n) \equiv 1-n \pmod * CDC(n) \equiv 2n+1 \pmod {{Num-stub Figurate numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Icosahedral Number
In mathematics, the centered icosahedral numbers also known as cuboctahedral numbers are a sequence of numbers, describing two different representations for these numbers as three-dimensional figurate numbers. As centered icosahedral numbers, they are centered numbers representing points arranged in the shape of a regular icosahedron. As cuboctahedral numbers, they represent points arranged in the shape of a cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ..., and are a magic number for the face-centered cubic lattice. The centered icosahedral number for a specific n is given by \frac. The first such numbers are References * . {{Num-stub Figurate numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stella Octangula Number
In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form .. The sequence of stella octangula numbers is :0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... Only two of these numbers are square. Ljunggren's equation There are only two positive square stella octangula numbers, and , corresponding to and respectively. The elliptic curve describing the square stella octangula numbers, :m^2 = n (2n^2 - 1) may be placed in the equivalent Weierstrass form :x^2 = y^3 - 2y by the change of variables , . Because the two factors and of the square number are relatively prime, they must each be squares themselves, and the second change of variables X=m/\sqrt and Y=\sqrt leads to Ljunggren's equation :X^2 = 2Y^4 - 1 A theorem of Siegel states that every elliptic curve has only finitely many integer solutions, and found a difficult proof that the only integer solutions to his equation were and , corresponding to the two square stel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
