Generator

Odd numbers as square shells

A generator in the geometry library, called 35 times across 8 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.

Numbers made of pyramids

The numbers that need five

Most large numbers need four

Ways to write a number, three against four

Each dimension costs more summands

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
the 1-th 5-gonal number is 1 dots, none repeated ×9
Fermat's bound of 3 is reached for 3-gonal numbers ×6
for power 2 the remainder approaches m/12 ×4
no number up to 20000 needs more than 4 4-gonal numbers (Fermat, Cauchy) ×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 a thousand to fifty thousand ×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
below a million exactly 241 numbers need five tetrahedral numbers, the last 343,867 ×1
between 3 and 12 steps ×1
each of the 54 represents every number up to 300, as the fifteen theorem promises ×1
every even number missed is 4^a(16b + 6), a family no form of this shape can reach ×1
every number an escalator misses first is one of 1, 2, 3, 5, 6, 7, 10, 14, 15 ×1
every number is a sum of three triangular numbers ×1
every odd Bernoulli number past the first is zero ×1
exactly 54 diagonal forms pass the fifteen test ×1
Faulhaber's polynomial gives the staircase's total ×1
four tetrahedral numbers give many more representations than three ×1
n is two triangular numbers exactly when 4n + 1 is a sum of two squares ×1
no number below ten million needs more than seven octahedral numbers ×1
only the centre cell is left over ×1
polygons with three to eight sides ×1
powers from 2 to 7 ×1
powers up to between 3 and 8 ×1
Ramanujan's fifty-fifth form, x² + 2y² + 5z² + 5w², misses 15 ×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, polydots, polycount, polytable, escalator, ramanujan54, lopsided ×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 odd numbers missed up to the range end at 2719, eighteen of them ×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
to two million the most needed is 3, 5 and 8 ×1
two to six shells ×1
up to 20000, only 9, 21, 31, 43, 55 and 89 need five pentagonal numbers ×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.

Geometry

Fifteen numbers decide every number

Lagrange proved that x² + y² + z² + w² takes every whole value. Ramanujan asked which other sums of four weighted squares do, and in 1917 listed fifty-five. One of them is wrong: x² + 2y² + 5z² + 5w² misses 15. The mistake points at a theorem found eighty years later — to know whether such a form reaches every number, it is enough to check that it reaches 1, 2, 3, 5, 6, 7, 10, 14 and 15. Nine checks decide infinitely many cases, and the nine numbers come out of an escalation any one can run.

Geometry

Five pyramids for every number

Stack balls in triangular pyramids and the piles hold 1, 4, 10, 20, 35, … balls. Frederick Pollock guessed in 1850 that every whole number is a sum of at most five such piles. Seventeen needs five — 10 + 4 + 1 + 1 + 1 — and so do 240 other numbers, the largest 343,867; past that, every number up to ten million needs four or fewer. Eight is proved to be enough. Five is not, and the reason is the same one that makes the three-dimensional cousins of the classical theorems hard.

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

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

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