Figurate number

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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 term can mean

Contents

Terminology

Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". [2]

In historical works about Greek mathematics the preferred term used to be figured number. [3] [4]

In a use going back to Jakob Bernoulli's Ars Conjectandi, [1] the term figurate number is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successive triangular numbers, etc. These turn out to be the binomial coefficients. In this usage the square numbers (4, 9, 16, 25, ...) would not be considered figurate numbers when viewed as arranged in a square.

A number of other sources use the term figurate number as synonymous for the polygonal numbers, either just the usual kind or both those and the centered polygonal numbers. [5]

History

The mathematical study of figurate numbers is said to have originated with Pythagoras, possibly based on Babylonian or Egyptian precursors. Generating whichever class of figurate numbers the Pythagoreans studied using gnomons is also attributed to Pythagoras. Unfortunately, there is no trustworthy source for these claims, because all surviving writings about the Pythagoreans [6] are from centuries later. [7] It seems to be certain that the fourth triangular number of ten objects, called tetractys in Greek, was a central part of the Pythagorean religion, along with several other figures also called tetractys.[ citation needed ] Figurate numbers were a concern of Pythagorean geometry.

The modern study of figurate numbers goes back to Pierre de Fermat, specifically the Fermat polygonal number theorem. Later, it became a significant topic for Euler, who gave an explicit formula for all triangular numbers that are also perfect squares, among many other discoveries relating to figurate numbers.

Figurate numbers have played a significant role in modern recreational mathematics. [8] In research mathematics, figurate numbers are studied by way of the Ehrhart polynomials, polynomials that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor. [9]

Triangular numbers and their analogs in higher dimensions

The triangular numbers for n = 1, 2, 3, ... are the result of the juxtaposition of the linear numbers (linear gnomons) for n = 1, 2, 3, ...:

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These are the binomial coefficients . This is the case r = 2 of the fact that the rth diagonal of Pascal's triangle for r ≥ 0 consists of the figurate numbers for the r-dimensional analogs of triangles (r-dimensional simplices).

The simplicial polytopic numbers for r = 1, 2, 3, 4, ... are:

The terms square number and cubic number derive from their geometric representation as a square or cube. The difference of two positive triangular numbers is a trapezoidal number.

Gnomon

The gnomon is the piece added to a figurate number to transform it to the next larger one.

For example, the gnomon of the square number is the odd number, of the general form 2n + 1, n = 0, 1, 2, 3, .... The square of size 8 composed of gnomons looks like this:


8   8   8   8   8   8   8   8
8   7   7   7   7   7   7   7
8   7   6   6   6   6   6   6
8   7   6   5   5   5   5   5
8   7   6   5   4   4   4   4
8   7   6   5   4   3   3   3
8   7   6   5   4   3   2   2
8   7   6   5   4   3   2   1

To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to the end of each row (n elements), one to the end of each column (n elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure.

This gnomonic technique also provides a mathematical proof that the sum of the first n odd numbers is n2; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 82.

Notes

  1. 1 2 Dickson, L. E., History of the Theory of Numbers
  2. Simpson, J. A.; Weiner, E. S. C., eds. (1992). The Compact Oxford English Dictionary (2nd ed.). Oxford, England: Clarendon Press. p. 587.Missing or empty |title= (help)
  3. Heath, T., A history of Greek Mathematics by
  4. Maziarz, E. A., Greek Mathematical Philosophy
  5. "Figurate Numbers". Mathigon. Retrieved 2019-02-06.
  6. Taylor, Thomas, The Theoretic Arithmetic of the Pythagoreans
  7. Boyer, Carl B.; Merzbach, Uta C., A History of Mathematics (Second ed.), p. 48
  8. Kraitchik, Maurice (2006), Mathematical Recreations (2nd revised ed.), Dover Books, ISBN   978-0-486-45358-3
  9. Beck, M.; De Loera, J. A.; Develin, M.; Pfeifle, J.; Stanley, R. P. (2005), "Coefficients and roots of Ehrhart polynomials", Integer points in polyhedra—geometry, number theory, algebra, optimization, Contemp. Math., 374, Providence, RI: Amer. Math. Soc., pp. 15–36, MR   2134759 .

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Binomial coefficient Positive integers that occur as coefficients in the binomial theorem

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In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example,

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

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 nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting at the 0th triangular number, is

In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

  1. A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant 1 is a monomial, being equal to the empty product and to x0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power xn of x, with n a positive integer. If several variables are considered, say, then each can be given an exponent, so that any monomial is of the form with non-negative integers.
  2. A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is 1. For example, in this interpretation and are monomials.

A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. Centered hexagonal numbers have practical applications in materials logistics management.

Pyramidal number

A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an r-sided polygon of points. The term usually refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides. It is possible to extend the pyramidal numbers to higher dimensions.

Tetrahedral number

A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number, Ten, is the sum of the first n triangular numbers, that is,

Square pyramidal number Number representing the number of stacked spheres in a square pyramid

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. The square pyramidal numbers can be used to count number of squares in an n × n grid, or acute triangles in 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.

In mathematics, an integer-valued polynomial is a polynomial whose value is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial

Pronic number

A pronic number is a number which is the product of two consecutive integers, that is, a number of the form 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.

Centered square number

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a 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 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.

Centered triangular number

A centeredtriangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for n is given by the formula

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer.

A nonagonal number is a figurate number that extends the concept of triangular and square numbers to the nonagon. However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the nth nonagonal number counts the number of dots in a pattern of n nested nonagons, all sharing a common corner, where the ith nonagon in the pattern has sides made of i dots spaced one unit apart from each other. The nonagonal number for n is given by the formula:

Pell number

In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

Squared triangular number Theorem that the sum of the first n cubes is the square of the nth triangular number

In number theory, the sum of the first n cubes is the square of the nth triangular number. That is,

Gnomon (figure)

In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger figure of the same shape.

In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged.

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