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{{Short description|Size of a geometric arrangement of points}}
{{Pascal triangle simplex numbers.svg|2=[[hyperpyramid]]al figurate numbers}}
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 {{mvar|r}}-dimensional regular [[geometry|geometric]] pattern of {{mvar|r}}-dimensional [[Ball (mathematics)|balls]] such as a polygonal number (for {{math|''r'' {{=}} 2}}) or a '''polyhedral number''' (for {{math|''r'' {{=}} 3}}).
* a member of the subset of the sets above containing only triangular numbers, [[pyramidal numbers]], and their analogs in other dimensions.<ref name="Dickson">{{cite book |author-link=Leonard Eugene Dickson |first=L. E. |last=Dickson |title=History of the Theory of Numbers |year=1919 |url=https://archive.org/details/historyoftheoryo02dick |access-date=2021-08-15 |page=3 |volume=2 |isbn=978-0-8284-0086-2 |url-access=subscription}}</ref>

== Terminology ==

Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number".<ref>{{cite encyclopedia |editor1-last=Simpson |editor1-first=J. A. |editor2-last=Weiner |editor2-first=E. S. C. |title=Figural number |encyclopedia=The Compact Oxford English Dictionary |edition=2nd |location=Oxford, England |publisher=Clarendon Press |date=1992 |page=587}}</ref>

In historical works about [[Greek mathematics]] the preferred term used to be ''figured number''.<ref>{{cite book |title=[[A History of Greek Mathematics]] |author-link=Thomas Little Heath |first=Sir Thomas |last=Heath |volume=1 |year=1921 |publisher=Oxford at the Clarendon Press}}</ref><ref>{{cite book |title=Greek Mathematical Philosophy |first1=Edward A. |last1=Maziarz |first2=Thomas |last2=Greenwood |year=1968 |publisher=Barnes & Noble Books |isbn=978-1-56619-954-4}}</ref>

In a use going back to [[Jacob Bernoulli]]'s [[Ars Conjectandi]],<ref name="Dickson" /> the term ''figurate number'' is used for [[triangular number]]s made up of successive [[integer]]s, [[tetrahedral number]]s made up of successive triangular numbers, etc. These turn out to be the [[binomial coefficient]]s. In this usage the [[square number]]s (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 number]]s, either just the usual kind or both those and the [[centered polygonal number]]s.<ref>{{cite web |url=https://mathigon.org/course/sequences/figurate#full |title=Figurate Numbers |website=Mathigon |access-date=2021-08-15}}</ref>

== 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 [[Gnomon (figure)|gnomons]] is also attributed to Pythagoras. Unfortunately, there is no trustworthy source for these claims, because all surviving writings about the Pythagoreans<ref>{{cite book |title=The Theoretic Arithmetic of the Pythagoreans |first=Thomas |last=Taylor |year=2006 |isbn=978-1-898910-29-9 |publisher=Prometheus Trust}}</ref> are from centuries later.<ref>{{cite book |title=A History of Mathematics |edition=Second |first1=Carl B. |last1=Boyer |first2=Uta C. |last2 = Merzbach |author2-link=Uta Merzbach |page=48 |year=1991}}</ref> [[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]].<ref>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</ref> Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. [[36 (number)|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 [[Square triangular number|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.<ref>{{cite book |last1=Kraitchik |first1=Maurice |title=Mathematical Recreations |edition=2nd revised |publisher=[[Dover Books]] |isbn=978-0-486-45358-3 |year=2006}}</ref> In research mathematics, figurate numbers are studied by way of the [[Ehrhart polynomial]]s, [[polynomial]]s that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor.<ref>{{cite book |last1=Beck |first1=M. |last2=De Loera |first2=J. A. |author2-link=Jesús A. De Loera |last3=Develin |first3=M. |last4=Pfeifle |first4=J. |last5=Stanley |first5=R. P. |author5-link=Richard P. Stanley |contribution=Coefficients and roots of Ehrhart polynomials |location=Providence, RI |mr=2134759 |pages=15–36 |publisher=Amer. Math. Soc. |series=Contemp. Math. |title=Integer points in polyhedra—geometry, number theory, algebra, optimization |volume=374 |year=2005}}</ref>

== Triangular numbers and their analogs in higher dimensions ==

The [[triangular number]]s for {{math|''n'' {{=}} 1, 2, 3, ...}} are the result of the juxtaposition of the linear numbers (linear gnomons) for {{math|''n'' {{=}} 1, 2, 3, ...}}:
{| cellpadding="10"
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|}

These are the binomial coefficients <math>\textstyle \binom{n+1}{2} </math>. This is the case {{math|''r'' {{=}} 2}} of the fact that the {{mvar|r}}th diagonal of [[Pascal's triangle#Patterns and properties|Pascal's triangle]] for {{math|''r'' ≥ 0}} consists of the figurate numbers for the {{mvar|r}}-dimensional analogs of triangles ({{mvar|r}}-dimensional [[simplex|simplices]]).

The simplicial polytopic numbers for {{math|''r'' {{=}} 1, 2, 3, 4, ...}} are:
*<math>P_1(n) = \frac{n}{1} = \binom{n+0}{1}=\binom{n}{1} </math> (linear numbers),
*<math>P_2(n) = \frac{n(n+1)}{2} = \binom{n+1}{2}</math> ([[triangular number]]s),
*<math>P_3(n) = \frac{n(n+1)(n+2)}{6} = \binom{n+2}{3}</math> ([[tetrahedral number]]s),
*<math>P_4(n) = \frac{n(n+1)(n+2)(n+3)}{24} = \binom{n+3}{4}</math> (pentachoric numbers, [[pentatopic number]]s, 4-simplex numbers),
<math>\qquad\vdots</math>
*<math>P_r(n) = \frac{n(n+1)(n+2)\cdots(n+r-1)}{r!} = \binom{n+(r-1)}{r}</math> ({{mvar|r}}-topic numbers, {{mvar|r}}-[[simplex]] numbers).

The terms ''[[square number]]'' and ''[[cubic number]]'' derive from their geometric representation as a [[Square (geometry)|square]] or [[cube (geometry)|cube]]. The difference of two positive triangular numbers is a [[trapezoidal number]].

== Gnomon ==
{{See also|Gnomon (figure)#Building figurate numbers}}
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 {{math|2''n'' + 1}}, {{math|''n'' {{=}} 0, 1, 2, 3, ...}}.  The square of size 8 composed of gnomons looks like this:

:<math>\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}</math>

To transform from the ''{{mvar|n}}-square'' (the square of size {{mvar|n}}) to the {{math|(''n'' + 1)}}-square, one adjoins {{math|2''n'' + 1}} elements: one to the end of each row ({{mvar|n}} elements), one to the end of each column ({{mvar|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 {{mvar|n}} odd numbers is {{math|''n''<sup>2</sup>}}; the figure illustrates {{nowrap|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15}} = 64 = 8<sup>2</sup>.

There is a similar '''gnomon''' with [[centered hexagonal number]]s adding up to make cubes of each integer number.

== Notes ==

{{reflist}}

== References ==
* {{Citation | last1=Gazalé | first1=Midhat J. | title=Gnomon: From Pharaohs to Fractals | journal=European Journal of Physics | publisher=[[Princeton University Press]] | isbn=978-0-691-00514-0 | year=1999| volume=20 | issue=6 | page=523 | doi=10.1088/0143-0807/20/6/501 | bibcode=1999EJPh...20..523G }}
* {{Citation | last1=Deza | first1=Elena|author1-link=Elena Deza | first2=Michel Marie | last2 = Deza | title=Figurate Numbers, First Edition | publisher=[[World Scientific]] | isbn=978-981-4355-48-3 | year=2012}}
* {{Citation | last1=Heath | first1=Thomas Little | title=A history of Greek Mathematics: Volume 1. From Thales to Euclid | publisher=[[Adamant Media Corporation]] | isbn=978-0-543-97448-8 | year=2000}}
* {{Citation | last1=Heath | first1=Thomas Little | title=A history of Greek Mathematics: Volume 2. From Aristarchus to Diophantus | publisher=[[Adamant Media Corporation]] | isbn=978-0-543-96877-7 | year=2000}}
* {{Citation | last1=Dickson | first1=Leonard Eugene | author-link = L. E. Dickson | title=[[History of the Theory of Numbers]] | publisher=[[Chelsea Publishing Company, Inc.|Chelsea Publishing Co]] | asin=B000OKO3TK | year=1923}}
* {{Citation | last1=Boyer | first1=Carl B. | first2=Uta C. | last2= Merzbach | title=A History of Mathematics | edition= 2nd }}

{{Figurate numbers}}
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[[Category:Figurate numbers| ]]
[[Category:Integer sequences]]