Square numbers
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
Completing the square, by completing a square
The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.
The square that cannot shrink
The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.
Three triangular numbers, and no fewer
On 10 July 1796 Gauss wrote in his diary: ΕΥΡΗΚΑ — num = Δ + Δ + Δ. Every whole number is a sum of three triangular numbers. Two are not enough, and not by a little: the numbers that are sums of two thin out to a share of nought. Both facts are statements about squares in disguise, and one picture translates them.
A box with every face diagonal whole
Pythagoras gives a rectangle with whole sides and a whole diagonal — 3, 4, 5. A box with whole edges and whole diagonals on all three faces is far rarer: the smallest is 44 × 117 × 240, found in 1719, and there are only 113 essentially different ones with every edge below 300,000. Ask for the long diagonal through the box to be whole as well and the answer is unknown. No such perfect box has ever been found, and nobody has proved there is none.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysTriangular numbersAlgebra tilesAreaCompleting the squareConjectureContinued fractionsDensityDescentDiophantine equationDissectionDivisibility