Concept

Gnomon

The piece that, added to a figure, produces a larger figure of the same shape: an L of 2n + 1 dots turns an n by n square into the next square. Adding gnomons one at a time is the oldest way of proving that sums of simple sequences are squares, triangles and other shapes.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Odd numbers as square shells. Nested L-shaped shells of 1, 3, 5 … 11 cells stack into a 6 by 6 square.

Every square is a stack of odd numbers

Add up the odd numbers in order and the running totals are 1, 4, 9, 16, 25. This is not a coincidence, and the reason fits in a single picture.

geometry · Figurate numbers

The fewest pentagonal numbers adding to each number up to 120. Fewest pentagonal numbers summing to 1..120; the most ever needed up to 20000 is 5, by 9, 21, 31, 43, 55, 89.

Six numbers that need five pentagons

In 1638 Fermat wrote that every whole number is a sum of three triangular numbers, four squares, five pentagonal numbers, six hexagonal numbers, and so on for every polygon — and that he had a proof he would not write down. The claim is true; Cauchy proved it in 1813. What the claim hides is how unequal the cases are. Triangles and squares need their full count infinitely often. For pentagons, only six numbers ever need all five — 9, 21, 31, 43, 55 and 89 — and from hexagons on, two apiece.

geometry · Figurate numbers

Named alongside it

The objects these essays reach for when they reach for this one.

Triangular numbers Counting two ways Exhaustive search Figurate numbers Quadratic form Square numbers Sum of squares Uniform acceleration

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