Figurate number

Last updated August 04, 2024

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

polygonal number
a number represented as a discrete $r$ -dimensional regular geometric pattern of $r$ -dimensional balls such as a polygonal number (for $r = 2$ ) or a polyhedral number (for $r = 3$ ).
a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions.^[1]

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 Jacob 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] Speusippus is the earliest source to expose the view that ten, as the fourth triangular number, was in fact the tetractys, supposed to be of great importance for Pythagoreanism.^[8] Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. 36 is a both a square and a triangle and also various rectangles.

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.^[9] 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.^[10]

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 $n = 1, 2, 3, ...$ :

These are the binomial coefficients $\textstyle {\binom {n+1}{2}}$ . This is the case $r = 2$ of the fact that the $r$ th diagonal of Pascal's triangle for $r \geq 0$ 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:

$P_{1}(n)={\frac {n}{1}}={\binom {n+0}{1}}={\binom {n}{1}}$ (linear numbers),
$P_{2}(n)={\frac {n(n+1)}{2}}={\binom {n+1}{2}}$ (triangular numbers),
$P_{3}(n)={\frac {n(n+1)(n+2)}{6}}={\binom {n+2}{3}}$ (tetrahedral numbers),
$P_{4}(n)={\frac {n(n+1)(n+2)(n+3)}{24}}={\binom {n+3}{4}}$ (pentachoric numbers, pentatopic numbers, 4-simplex numbers),

$\qquad \vdots$

$P_{r}(n)={\frac {n(n+1)(n+2)\cdots (n+r-1)}{r!}}={\binom {n+(r-1)}{r}}$ ( $r$ -topic numbers, $r$ -simplex numbers).

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 $2 n + 1$ , $n = 0, 1, 2, 3, ...$ . The square of size 8 composed of gnomons looks like this:

{\begin{matrix}1&2&3&4&5&6&7&8\\2&2&3&4&5&6&7&8\\3&3&3&4&5&6&7&8\\4&4&4&4&5&6&7&8\\5&5&5&5&5&6&7&8\\6&6&6&6&6&6&7&8\\7&7&7&7&7&7&7&8\\8&8&8&8&8&8&8&8\end{matrix}}

To transform from the $n$ -square (the square of size $n$ ) to the $(n + 1)$ -square, one adjoins $2 n + 1$ 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 $n 2$ ; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8².

There is a similar gnomon with centered hexagonal numbers adding up to make cubes of each integer number.

Notes

1 2 Dickson, L. E. (1919). History of the Theory of Numbers . Vol. 2. p. 3. ISBN 978-0-8284-0086-2 . Retrieved 2021-08-15.
↑ Simpson, J. A.; Weiner, E. S. C., eds. (1992). "Figural number". The Compact Oxford English Dictionary (2nd ed.). Oxford, England: Clarendon Press. p. 587.
↑ Heath, Sir Thomas (1921). A History of Greek Mathematics . Vol. 1. Oxford at the Clarendon Press.
↑ Maziarz, Edward A.; Greenwood, Thomas (1968). Greek Mathematical Philosophy. Barnes & Noble Books. ISBN 978-1-56619-954-4.
↑ "Figurate Numbers". Mathigon. Retrieved 2021-08-15.
↑ Taylor, Thomas (2006). The Theoretic Arithmetic of the Pythagoreans. Prometheus Trust. ISBN 978-1-898910-29-9.
↑ Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (Second ed.). p. 48.
↑ Zhmud, Leonid (2019): From Number Symbolism to Arithmology. In: L. Schimmelpfennig (ed.): Number and Letter Systems in the Service of Religious Education. Tübingen: Seraphim, 2019. p.25-45
↑ Kraitchik, Maurice (2006). Mathematical Recreations (2nd revised ed.). Dover Books. ISBN 978-0-486-45358-3.
↑ 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. Vol. 374. Providence, RI: Amer. Math. Soc. pp. 15–36. MR 2134759.

Related Research Articles

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 $n \geq k \geq 0$ and is written $It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n; this coefficient can be computed by the multiplicative formula$

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

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

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 $32$ and can be written as $3 \times 3$ .

In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers.

In mathematics and combinatorics, 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. The following figures illustrate this arrangement for the first four centered hexagonal numbers:

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number p_n is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.

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.

<span class="mw-page-title-main">Tetrahedral number</span> Polyhedral number representing a tetrahedron

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 $n$ th tetrahedral number, $Te n$ , is the sum of the first $n$ triangular numbers, that is,

In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the 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.

A pronic number is a number that is the product of two consecutive integers, that is, a number of the form $. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.$

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.

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

A nonagonal number, or an enneagonal 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 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:

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 $i 2$ points on the square faces of the $i$ th layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has $n + 1$ points along each of its edges.

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.

288 is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".

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.

In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers $as a polynomial in n . In modern notation, Faulhaber's formula is Here, is the binomial coefficient " p + 1 choose r ", and the B j are the Bernoulli numbers with the convention that .$

<span class="mw-page-title-main">Integer triangle</span> Triangle with integer side lengths

An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles.

References

Gazalé, Midhat J. (1999), "Gnomon: From Pharaohs to Fractals", European Journal of Physics, 20 (6), Princeton University Press: 523, Bibcode:1999EJPh...20..523G, doi:10.1088/0143-0807/20/6/501, ISBN 978-0-691-00514-0
Deza, Elena; Deza, Michel Marie (2012), Figurate Numbers, First Edition, World Scientific, ISBN 978-981-4355-48-3
Heath, Thomas Little (2000), A history of Greek Mathematics: Volume 1. From Thales to Euclid, Adamant Media Corporation, ISBN 978-0-543-97448-8
Heath, Thomas Little (2000), A history of Greek Mathematics: Volume 2. From Aristarchus to Diophantus, Adamant Media Corporation, ISBN 978-0-543-96877-7
Dickson, Leonard Eugene (1923), History of the Theory of Numbers , Chelsea Publishing Co, ASIN B000OKO3TK
Boyer, Carl B.; Merzbach, Uta C., A History of Mathematics (2nd ed.)

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[Dickson-1] 1 2 Dickson, L. E. (1919). History of the Theory of Numbers . Vol. 2. p. 3. ISBN 978-0-8284-0086-2 . Retrieved 2021-08-15.

[2] Simpson, J. A.; Weiner, E. S. C., eds. (1992). "Figural number". The Compact Oxford English Dictionary (2nd ed.). Oxford, England: Clarendon Press. p. 587.

[3] Heath, Sir Thomas (1921). A History of Greek Mathematics . Vol. 1. Oxford at the Clarendon Press.

[4] Maziarz, Edward A.; Greenwood, Thomas (1968). Greek Mathematical Philosophy. Barnes & Noble Books. ISBN 978-1-56619-954-4.

[5] "Figurate Numbers". Mathigon. Retrieved 2021-08-15.

[6] Taylor, Thomas (2006). The Theoretic Arithmetic of the Pythagoreans. Prometheus Trust. ISBN 978-1-898910-29-9.

[7] Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (Second ed.). p. 48.

[8] Zhmud, Leonid (2019): From Number Symbolism to Arithmology. In: L. Schimmelpfennig (ed.): Number and Letter Systems in the Service of Religious Education. Tübingen: Seraphim, 2019. p.25-45

[9] Kraitchik, Maurice (2006). Mathematical Recreations (2nd revised ed.). Dover Books. ISBN 978-0-486-45358-3.

[10] 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. Vol. 374. Providence, RI: Amer. Math. Soc. pp. 15–36. MR 2134759.

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