Figurate number

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

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

In historical works about Greek mathematics the preferred term used to be figured number.

In a use going back to Jakob Bernoulli's Ars Conjectandi, 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.

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 are from centuries later. 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. Figurate numbers were a concern of Pythagorean geometry.

The modern study of figurate numbers goes back to 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.

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