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Concept
Figurate number
Summary
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.
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 Jacob 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. 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. Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g.
Official source
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