|
Octahedral Number
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahedral numbers are: :1 (number), 1, 6 (number), 6, 19 (number), 19, 44 (number), 44, 85 (number), 85, 146, 231, 344, 489, 670, 891 . Properties and applications The octahedral numbers have a generating function : \frac = \sum_^ O_n z^n = z +6z^2 + 19z^3 + \cdots . Sir Frederick Pollock, 1st Baronet, Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers. This statement, the Pollock octahedral numbers conjecture, has been proven true for all but finitely many numbers. In chemistry, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called Magic number (chemistry), magic numbers.. Relation to other figurate numbers Square p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Pyramidal Number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid (geometry), pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first n positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number. History The pyramidal numbers were one of the few types of three-dimensional fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular Polyhedron
In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). These polyhedra include: *The thirteen Archimedean solids. ** The elongated square gyrobicupola (also called a pseudo-rhombicuboctahedron), a Johnson solid, has identical vertex figures (3.4.4.4) but because of a twist it is not vertex-transitive. Branko Grünbaum argued for including it as a 14th Archimedean solid. *An infinite series of convex prisms. *An infinite series of convex antiprisms (their semiregular nature was first observed by Kepler). These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the ''Timaeus (dialogue), Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramidal Number
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions. Formula The formula for the th -gonal pyramidal number is :P_n^r= \frac, where , . This formula can be factored: :P_n^r=\frac=\left(\frac\right)\left(\frac\right)=T_n \cdot \frac, where is the th triangular number. Sequences The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are: : 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... The first few square pyramidal numbers are: : 1, 5, 14, 30, 55, 91, 140, 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Number
In mathematics, a polygonal number is a Integer, number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of Pronic number, oblong, Triangular Number, triangular, and Square number, square numbers. Definition and examples The number 10 for example, can be arranged as a triangle (see triangular number): : But 10 cannot be arranged as a square (geometry), square. The number 9, on the other hand, can be (see square number): : Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number): : By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Faulhaber
Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician, specifically, a calculator ('':de:Rechenmeister, Rechenmeister''). Biography Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. In 1620, while in Ulm, René Descartes, Descartes probably corresponded with Faulhaber to discuss algebraic solutions of polynomial equations. He worked as fortification engineer in various cities (notably Basel, where he was fortification engineer from 1622 to 1624, and Frankfurt), and also worked under Maurice, Prince of Orange in the Netherlands. He also built water wheels in his home town and geometrical instruments for the military. Faulhaber made the first publication of Henry Briggs (mathematician), Henry Briggs's Logarithm in Germany. He is also credited with the first printed solution of equal temperament. He died in Ulm. Faulhaber's major contribution w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramount to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, and later becoming a central intellectual of the Dutch Golden Age. Although he served a Dutch Reformed Church, Protestant state and was later counted as a Deism, deist by critics, Descartes was Roman Catholicism, Roman Catholic. Many elements of Descartes's philosophy have precedents in late Aristotelianism, the Neostoicism, revived Stoicism of the 16th century, or in earlier philosophers like Augustine of Hippo, Augustine. In his natural philosophy, he differed from the Scholasticism, schools on two major point ... [...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] |
Francesco Maurolico
Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian language, Italian: ''Francesco Maurolico''; ; Sicilian language, Sicilian: ''Francescu Maurolicu''; 16 September 1494 – 22 July 1575) was an Italian mathematician and astronomer from the Kingdom of Sicily. He made contributions to the fields of geometry, optics, conics, mechanics, music, and astronomy. He edited the works of classical authors including Archimedes, Apollonius of Perga, Apollonius, Autolycus of Pitane, Autolycus, Theodosius of Bithynia, Theodosius and Serenus of Antinouplis, Serenus. He also composed his own unique treatises on mathematics and mathematical science. Life Francesco was born in Messina with the surname of Marulì, although the surname is sometimes reported as "Mauroli". He was one of seven sons of Antonio Marulì, a government official, and Penuccia. His father was a Greek physician who fled Constantinople when the Ottomans invaded the city. Antonio had studied with the Neoplatonic He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Square Number
In elementary number theory, a centered square number is a Centered polygonal number, centered figurate number that gives the number of dots in a Square (geometry), square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given Taxicab geometry, city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties. The figures for the first four centered square numbers are shown below: : Each centered square number is the sum of successive squares. Example: as shown in the following figure of Floyd's triangle, 25 is a centered square number, and is the sum of the square 16 (yellow rhombus formed by shearing a square) and of the next smaller sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
