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The triangular numbers can be generalized in various ways. As used here, and in a number other places, the term figurate number is meant to be broadly inclusive.
In historical works about Greek mathematics the preferred term is figured number. For example A history of Greek Mathematics by T. Heath and Greek Mathematical Philosophy by E.A. Maziarz.
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. In other words 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. This is the sense in which L.E. Dickson uses the term in his History of the Theory of Numbers.
The first few triangular numbers can be built from rows of 1, 2, 3, 4, 5, and 6 items:
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These are seen to be the binomial coefficients . This is the case r=2 of the fact that the rth diagonal of Pascals Triangle for consists of the figurate numbers for the r-dimensional analogs of triangles (simplices).
Polytopic numbers for r = 2, 3, and 4 are:
Figurate numbers were a concern of Pythagorean geometry, since Pythagoras is credited with initiating them, and the notion that these numbers are generated from a gnomon or basic unit. The gnomon is the piece added to a figurate number to transform it to the next bigger one.
For example, the gnomon of the square number is the odd number, of the general form 2n + 1, n = 0, 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this:
8 8 8 8 8 8 8 8
8 7 7 7 7 7 7 7
8 7 6 6 6 6 6 6
8 7 6 5 5 5 5 5
8 7 6 5 4 4 4 4
8 7 6 5 4 3 3 3
8 7 6 5 4 3 2 2
8 7 6 5 4 3 2 1
To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 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.
Note that this gnomonic technique also provides a proof that the sum of the first n odd numbers is n2; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 82.
Demonstration of mathematical properties
School children construct figurate numbers from pebbles, bottle caps, etc. As a bonus, children can use figurate numbers to discover the commutative law and associative law for addition and multiplication — laws usually dictated to them — by building rows and tables of dots.
For example, the additive commutativity of 2 + 3 = 3 + 2 = 5 becomes:
And the multiplicative commutativity of 2 × 3 = 3 × 2 = 6 becomes:
Besides the subtractive method, the additive method can also approximate square roots of positive integers and solve quadratic equations.
The concepts of figurate numbers and gnomon implicitly anticipate the modern concept of recursion.