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In mathematics, a pyramid number, or square pyramidal number, is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
that counts the number of stacked spheres in a
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
with a square base. The study of these numbers goes back to
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
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 number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
s, 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 triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's an ...
s 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 numbers studied in
Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
, in works by
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works '' Introduction to Arithmetic'' and '' Manual of Harmonics'' in Greek. He was bo ...
, Theon of Smyrna, and
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer o ...
. Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
, who used this sum as a
lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), ...
as part of a study of the volume of a cone, and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares. The square pyramidal numbers were also one of the families of figurate numbers studied by
Japanese mathematicians Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese dia ...
of the wasan period, who named them "kirei saijō suida" (with modern
kanji are the logographic Chinese characters taken from the Chinese script and used in the writing of Japanese. They were made a major part of the Japanese writing system during the time of Old Japanese and are still used, along with the subsequ ...
, 奇零 再乗 蓑深). The same problem, formulated as one of counting the
cannonball A round shot (also called solid shot or simply ball) is a solid spherical projectile without explosive charge, launched from a gun. Its diameter is slightly less than the bore of the barrel from which it is shot. A round shot fired from a lar ...
s in a square pyramid, was posed by
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 rebelli ...
to mathematician Thomas Harriot in the late 1500s, while both were on a sea voyage. The cannonball problem, asking whether there are any square pyramidal numbers that are also square numbers other than 1 and 4900, is said to have developed out of this exchange. Édouard Lucas found the 4900-ball pyramid with a square number of balls, and in making the cannonball problem more widely known, suggested that it was the only nontrivial solution. After incomplete proofs by Lucas and Claude-Séraphin Moret-Blanc, the first complete proof that no other such numbers exist was given by
G. N. Watson George Neville Watson (31 January 1886 – 2 February 1965) was an English mathematician, who applied complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. Whittaker's ''A Course of Mode ...
in 1918.


Formula

If spheres are packed into square pyramids whose number of layers is 1, 2, 3, etc., then the square pyramidal numbers giving the numbers of spheres in each pyramid are: These numbers can be calculated algebraically, as follows. If a pyramid of spheres is decomposed into its square layers with a square number of spheres in each, then the total number P_n of spheres can be counted as the sum of the number of spheres in each square, P_n = \sum_^nk^2 = 1 + 4 + 9 + \cdots + n^2, and this
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, ma ...
can be solved to give a cubic polynomial, which can be written in several equivalent ways: P_n= \frac = \frac = \frac + \frac + \frac. This equation for a sum of squares is a special case of Faulhaber's formula for sums of powers, and may be proved by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
. More generally, figurate numbers count the numbers of geometric points arranged in regular patterns within certain shapes. The centers of the spheres in a pyramid of spheres form one of these patterns, but for many other types of figurate numbers it does not make sense to think of the points as being centers of spheres. In modern mathematics, related problems of counting points in integer polyhedra are formalized by the Ehrhart polynomials. These differ from figurate numbers in that, for Ehrhart polynomials, the points are always arranged in an integer lattice rather than having an arrangement that is more carefully fitted to the shape in question, and the shape they fit into is a polyhedron with lattice points as its vertices. Specifically, the Ehrhart polynomial of an integer polyhedron is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
that counts the number of integer points in a copy of that is expanded by multiplying all its coordinates by the number . The usual symmetric form of a square pyramid, with a
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordin ...
as its base, is not an integer polyhedron, because the topmost point of the pyramid, its apex, is not an integer point. Instead, the Ehrhart polynomial can be applied to an asymmetric square pyramid with a unit square base and an apex that can be any integer point one unit above the base plane. For this choice of , the Ehrhart polynomial of a pyramid is .


Geometric enumeration

As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems. For example, a common mathematical puzzle involves finding the number of squares in a large by square grid. This number can be derived as follows: *The number of squares found in the grid is . *The number of squares found in the grid is . These can be counted by counting all of the possible upper-left corners of squares. *The number of squares found in the grid is . These can be counted by counting all of the possible upper-left corners of squares. It follows that the number of squares in an square grid is: n^2 + (n-1)^2 + (n-2)^2 + (n-3)^2 + \ldots = \frac. That is, the solution to the puzzle is given by the th square pyramidal number. The number of rectangles in a square grid is given by the squared triangular numbers. The square pyramidal number P_n also counts the number of
acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's an ...
s formed from the vertices of a (2n+1)-sided regular polygon. For instance, an equilateral triangle contains only one acute triangle (itself), a regular
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...
has five acute
golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...
s within it, a regular heptagon has 14 acute triangles of two shapes, etc. More abstractly, when permutations of the rows or columns of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
are considered as equivalent, the number of 2\times 2 matrices with non-negative integer coefficients summing to n, for odd values of n, is a square pyramidal number.


Relations to other figurate numbers

The cannonball problem asks for the sizes of pyramids of cannonballs that can also be spread out to form a square array, or equivalently, which numbers are both square and square pyramidal. Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number. The square pyramidal numbers can be expressed as sums of
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s: P_n = \binom + \binom. The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
s. If a tetrahedron is reflected across one of its faces, the two copies form a
triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, ...
. The square pyramidal numbers are also the figurate numbers of the triangular bipyramids, and this formula can be interpreted as an equality between the square pyramidal numbers and the triangular bipyramidal numbers. Analogously, reflecting a square pyramid across its base produces an octahedron, from which it follows that each octahedral number is the sum of two consecutive square pyramidal numbers. Square pyramidal numbers are also related to tetrahedral numbers in a different way: the points from four copies of the same square pyramid can be rearranged to form a single tetrahedron with twice as many points along each edge. That is, 4P_n=Te_=\binom. To see this, arrange each square pyramid so that each layer is directly above the previous layer, e.g. the heights are
4321
3321
2221
1111
Four of these can then be joined by the height pillar to make an even square pyramid, with layers 4, 16, 36, \dots. Each layer is the sum of consecutive triangular numbers, i.e. (1+3), (6+10), (15+21), \dots, which, when totalled, sum to the tetrahedral number.


Other properties

The alternating series of unit fractions with the square pyramidal numbers as denominators is closely related to the Leibniz formula for , although it converges more quickly. It is: \begin \sum_^& (-1)^\frac\\ &=1-\frac+\frac-\frac+\frac-\frac+\frac-\frac+\cdots\\ &=6(\pi-3)\\ &\approx 0.849556.\\ \end In
approximation theory In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...
, the sequences of odd numbers, sums of odd numbers (square numbers), sums of square numbers (square pyramidal numbers), etc., form the coefficients in a method for converting Chebyshev approximations into
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s.


References


External links

* {{Classes of natural numbers Figurate numbers Pyramids Articles containing video clips