Generator

Odd numbers as square shells

A generator in the geometry library, called 19 times across 5 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

figurate is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

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

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the polynomial for power 1 gives the exact sum at n = 1 ×150
0 and 3: triangular triples match odd-square triples ×121
for power 2 the remainder approaches m/12 ×4
the first 6 odd numbers sum to 6² ×4
a range from 60 to 400 ×2
a number from 1 to 300 ×1
a range from ten thousand to two million ×1
a triangular index from 1 to 7 ×1
an even index from 10 to 40 ×1
and it is the centre ×1
between 3 and 12 steps ×1
every number is a sum of three triangular numbers ×1
every odd Bernoulli number past the first is zero ×1
Faulhaber's polynomial gives the staircase's total ×1
n is two triangular numbers exactly when 4n + 1 is a sum of two squares ×1
only the centre cell is left over ×1
powers from 2 to 7 ×1
powers up to between 3 and 8 ×1
share × √(log x) changes little over two decades ×1
the kind is gnomon or one of triangular, galileo, staircase, faulhaber, correction, bernoulli, eight, trisum, twotri, twotridensity, threetri ×1
the L-shaped shells fill the square ×1
the leading coefficient is 1/(m+1), the integral's ×1
the next is 1/2, half the last step ×1
the power is between 1 and 6 ×1
the share keeps drifting down ×1
the signs alternate, positive at 2, 6, 10, … ×1
the size matches 2(2k)!/(2π)^(2k) ×1
the staircase stands above the curve by about half its last step ×1
the three triangles add to the number ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Analysis

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

Number

Numbers that are their own parts

Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.

Geometry

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

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

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.

The whole library · What the figures prove