Matpic
mathematics, in pictures
Concept

Triangular numbers

The counts of dots in a triangular arrangement - 1, 3, 6, 10 and onward - each the running total of the whole numbers up to its own index. They are the third diagonal of Pascal's triangle, because a diagonal there is the running total of the one before it, and twice one of them is a rectangle.

Named by 5 essays across 2 fields — 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 A run down a diagonal, and the entry it adds to. 9 rows of Pascal's triangle with 5 entries shaded and the entry they add to marked. The claim is checked by adding the shaded entries: 1 + 3 + 6 + 10 + 15 = 35.

The run that lands one place along

Add up a run of entries down one of Pascal's diagonals and the total is another entry of the triangle — one row further down and one place along. The same triangle holds four more sums of that kind, and each is a different question answered by the same additive rule.

discrete · Pascals triangle The sum of the first 6 squares, as a staircase over a curve. Bars of height k^2 for k from 1 to 6, totalling 91, drawn over the curve y = x^2, whose area up to 6 is 72.00. The slivers between staircase and curve hold 19.00, close to half the last bar.

Sums of powers, read off a staircase

Add the first n squares, or cubes, or seventh powers, and the answer is always a polynomial in n. Its first term is the area under a curve, its second is half of the last step, and every term after that is a correction for the corners of a staircase — which is where the Bernoulli numbers come from, and why they eventually grow without bound.

geometry · Figurate numbers 100 as three triangular numbers. 100 drawn as three triangles of dots with 36, 36 and 28 dots. There are 6 such decompositions.

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.

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.

Counting two waysFigurate numbersGnomonQuadratic formSquare numbersAlgebraic identityApproximationAsymptotic seriesBinomial coefficientCombinatorial proofDensityError term

All concepts