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Square Pyramidal Number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a 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 figurate num ... [...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 number of stacked spheres in a 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 figurate num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the ''lateral surface''; if the lateral surface is unbounded, it is a conical surface. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
55 (number)
55 (fifty-five) is the natural number following 54 (number), 54 and preceding 56 (number), 56. Mathematics 55 is *a triangular number (the sum of the consecutive numbers 1 to 10), and a doubly triangular number. *the 10th Fibonacci number. It is the largest Fibonacci number to also be a triangular number. * a square pyramidal number (the sum of the Square (algebra), squares of the integers 1 to 5) as well as a heptagonal number, and a centered nonagonal number. *In base 10, it is a Kaprekar number. Science *The atomic number of caesium. Astronomy *Messier object Messier 55, M55, a magnitude 7.0 globular cluster in the constellation Sagittarius (constellation), Sagittarius *The New General Catalogue object NGC 55, a magnitude 7.9 barred spiral galaxy in the constellation Sculptor (constellation), Sculptor Music * The name of a song by Kasabian. The song was released as a B side to ''Club Foot'' and was recorded live when the band performed at London's Brixton Academy. * " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
30 (number)
30 (thirty) is the natural number following 29 and preceding 31. In mathematics 30 is an even, composite, pronic number. With 2, 3, and 5 as its prime factors, it is a regular number and the first sphenic number, the smallest of the form , where is a prime greater than 3. It has an aliquot sum of 42, which is the second sphenic number. It is also: * A semiperfect number, since adding some subsets of its divisors (e.g., 5, 10 and 15) equals 30. * A primorial. * A Harshad number in decimal. * Divisible by the number of prime numbers ( 10) below it. * The largest number such that all coprimes smaller than itself, except for 1, are prime. * The sum of the first four squares, making it a square pyramidal number. * The number of vertices in the Tutte–Coxeter graph. * The measure of the central angle and exterior angle of a dodecagon, which is the petrie polygon of the 24-cell. * The number of sides of a triacontagon, which in turn is the petrie polygon of the 120-cell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
14 (number)
14 (fourteen) is a natural number following 13 and preceding 15. In relation to the word "four" ( 4), 14 is spelled "fourteen". In mathematics * 14 is a composite number. * 14 is a square pyramidal number. * 14 is a stella octangula number. * In hexadecimal, fourteen is represented as E * Fourteen is the lowest even ''n'' for which the equation φ(''x'') = ''n'' has no solution, making it the first even nontotient (see Euler's totient function). * Take a set of real numbers and apply the closure and complement operations to it in any possible sequence. At most 14 distinct sets can be generated in this way. ** This holds even if the reals are replaced by a more general topological space. See Kuratowski's closure-complement problem * 14 is a Catalan number. * Fourteen is a Companion Pell number. * According to the Shapiro inequality 14 is the least number ''n'' such that there exist ''x'', ''x'', ..., ''x'' such that :\sum_^ \frac < \frac where ''x'' = ''x'', ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 (number)
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has attained significance throughout history in part because typical humans have five digits on each hand. In mathematics 5 is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. Notably, 5 is equal to the sum of the ''only'' consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, ( 3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Édouard Lucas
__NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Lucas was born in Amiens and educated at the École Normale Supérieure. He worked in the Paris Observatory and later became a professor of mathematics at the Lycée Saint Louis and the Lycée Charlemagne in Paris. Lucas served as an artillery officer in the French Army during the Franco-Prussian War of 1870–1871. In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation: :\sum_^ n^2 = M^2\; with ''N'' > 1 is when ''N'' = 24 and ''M'' = 70. This is known as the cannonball problem, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a square pyramid out of them. It was not until 1918 that a proof (using elliptic functions) was found for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cannonball Problem
In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1. Formulation as a Diophantine equation When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their expedition to America. Édouard Lucas formulated the cannonball problem as a Diophantine equation :\sum_^ n^2 = M^2 or :\frac N(N+1)(2N+1) = \frac = M^2. Solution Lucas conjectured that the only solutions are ''N'' = 1, ''M'' = 1, and ''N'' = 24, ''M'' = 70, using either 1 or 4900 cannon balls. It was not until 1918 that G. N. Watson found a proof fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Harriot
Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his contributions in navigational techniques, working closely with John White to create advanced maps for navigation. While Harriot worked extensively on numerous papers on the subjects of astronomy, mathematics and navigation, he remains obscure because he published little of it, namely only ''The Briefe and True Report of the New Found Land of Virginia'' (1588). This book includes descriptions of English settlements and financial issues in Virginia at the time. He is sometimes credited with the introduction of the potato to the British Isles. Harriot was the first person to make a drawing of the Moon through a telescope, on 5 August 1609, about four months before Galileo Galilei. After graduating from St Mary Hall, Oxford, Harriot traveled to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Raleigh
Sir Walter Raleigh (; – 29 October 1618) was an English statesman, soldier, writer and explorer. One of the most notable figures of the Elizabethan era, he played a leading part in English colonisation of North America, suppressed rebellion in Ireland, helped defend England against the Spanish Armada and held political positions under Elizabeth I. Raleigh was born to a Protestant family in Devon, the son of Walter Raleigh and Catherine Champernowne. He was the younger half-brother of Sir Humphrey Gilbert and a cousin of Sir Richard Grenville. Little is known of his early life, though in his late teens he spent some time in France taking part in the religious civil wars. In his 20s he took part in the suppression of rebellion in the colonisation of Ireland; he also participated in the siege of Smerwick. Later, he became a landlord of property in Ireland and mayor of Youghal in East Munster, where his house still stands in Myrtle Grove. He rose rapidly in the favour of Quee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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