Editing Figurate number (section)

== 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>{{citation |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>{{citation |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>{{citation |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>{{citation |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>