Generator

The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3)

A generator in the computation library, called 55 times across 15 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.

degree 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

The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

Every rational number that could be a root of x³ − 2

The three-dimensional field a cube root of 2 generates

Looking for a polynomial with π as a root

Two cubics broken apart modulo the primes up to 60

The subgroups of a polynomial's symmetries, against the fields they name

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 linear factors of x³ − 2 mod 5 are its roots ×15
the linear factors of x³ − 3x + 1 mod 5 are its roots ×15
φ^0 is 1 + 0φ ×10
φ^1 has norm ±1 ×9
the highest power is a whole number between 5 and 12 ×2
√2 + ∛3 has degree six ×1
∛2 × ∛2 lands on a multiple of ∛2² ×1
∛2 × ∛2² lands on a multiple of 1 ×1
∛2 × 1 lands on a multiple of ∛2 ×1
∛2² × ∛2 lands on a multiple of 1 ×1
∛2² × ∛2² lands on a multiple of ∛2 ×1
∛2² × 1 lands on a multiple of ∛2² ×1
1 × ∛2 lands on a multiple of ∛2 ×1
1 × ∛2² lands on a multiple of ∛2² ×1
1 × 1 lands on a multiple of 1 ×1
a bigger subgroup names a smaller field, never the other way round ×1
a rational is a whole numerator over a non-zero whole denominator ×1
a subgroup of order two fixes exactly one root ×1
all three roots are real ×1
and grow without bound ×1
and the product's at √2 · φ ×1
and the sign alternates ×1
at least one of 2, 3 and 6 is a square modulo every prime ×1
both are monic ×1
divisors are taken of a positive whole number ×1
each ±√2 + ω^j∛3 is a root of the polynomial ×1
each number under a root is a whole number between 2 and 40 ×1
each number under a root is square-free, or the step is not a real step ×1
each pattern's share is its cycle type's share of the group ×1
each trace step divides exactly, as it must for an integer matrix ×1
every coefficient is a whole number under 400 ×1
every degree divides the degree of the field, six ×1
every pattern that occurs is a cycle type of the Galois group ×1
every positive unit is a power of ∛2 − 1 ×1
every product of basis elements is a multiple of a basis element ×1
every rational root found by sweeping is one the theorem listed ×1
every unit found is ± a whole-number combination of the two ×1
for p ≡ 2 mod 3, x³ − 2 has exactly one root ×1
it splits completely exactly when 2 and 3 are both squares ×1
no integer polynomial in the searched range vanishes at π ×1
no power of two is a multiple of three ×1
no whole number the rational-root theorem allows is a root of x³ − m ×1
some combination has a smaller degree than the bound ×1
the cell of ℤ[√5] has area 2√5, twice as large ×1
the cell of ℤ[φ] has area √5 ×1
the characteristic polynomial of the sum's matrix vanishes at √2 + φ ×1
the closest miss is a genuine miss ×1
the coefficient bound is a whole number between 2 and 8 ×1
the coefficient range is a whole number between 2 and 6 ×1
the control polynomial really vanishes at its algebraic number ×1
the cube root really cubes to it ×1
the degree is at most the product of the two degrees ×1
the degree of a tower of square roots is a power of two ×1
the degree of the tower is the size of its basis ×1
the denominators of the non-integer's powers never shrink ×1
the earlier powers are independent, so the solve is unique ×1
the exact value and the decimal one agree ×1
the factor degrees add to the degree ×1
the field is one the family knows ×1
the grid half-width is a whole number between 3 and 8 ×1
the group of six permutations has exactly six subgroups ×1
the largest degree searched is a whole number between 1 and 4 ×1
the largest prime listed is a whole number between 30 and 120 ×1
the leading coefficient is not zero ×1
the monic test agrees with a and b having the same parity ×1
the number being factorised is a whole number between 1 and 1000000 ×1
the number being searched for is one of pi, sqrt2, cbrt2, phi ×1
the number whose cube root is taken is between 2 and 30 ×1
the polynomial found vanishes at √2 + ∛3 ×1
the polynomial has degree between one and four ×1
the polynomial is named for the caption ×1
the polynomial vanishes at the number ×1
the powers of two are checked out to between 6 and 20 steps ×1
the product is the coefficient times the basis element it lands on ×1
the range of primes is a whole number between 2000 and 50000 ×1
the rank is never negative ×1
the same search does find a polynomial for an algebraic number ×1
the search bound is a whole number between 2 and 7 ×1
the searched number is a root of one of them ×1
the shares settle on 1/3 and 1/6 ×1
the size of the subgroup times the degree of its field is the degree of the whole ×1
the three logarithms of a unit add to zero, because its norm is ±1 ×1
the tower has one basis element per subset of its roots ×1
the tower is built from one, two or three square roots ×1
the two chosen units are independent ×1
the view is one the family draws ×1
x³ − 3x + 1 splits exactly when p ≡ ±1 mod 9 ×1
x⁴ − 10x² + 1 is never irreducible modulo a prime ×1
φ^k lies in ℤ[√5] exactly when k is a multiple of 3 ×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.

Algebra

A shared root, found without finding it

Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.

Algebra

A tower whose degrees multiply

Treat a field containing another as a vector space over it, and the size of an extension becomes a dimension — one that multiplies along a tower, so that three impossible constructions become arithmetic about which numbers divide which.

Computation

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

Algebra

How a polynomial breaks modulo the primes

Reduce x³ − 2 modulo a prime and it factors: into three linear pieces for some primes, one linear and one quadratic for others, not at all for the rest. Over the primes up to twenty thousand those three patterns occur a sixth, a half and a third of the time — exactly the shares of the identity, the flips and the rotations in the symmetry group of a triangle, the group that permutes the three cube roots of 2. A polynomial's factorisations modulo primes are a census of its Galois group.

Algebra

Seven powers in a space of six

Is √2 + ∛3 a root of some polynomial with whole-number coefficients? It lives in a field of dimension six, so its first seven powers are seven vectors in a six-dimensional space and must be dependent — and the dependency, solved exactly, is the polynomial. The same count shows every sum, product and quotient of algebraic numbers is algebraic, without ever needing a formula.

Computation

The angle that will not divide by three

Halving an angle costs one circle. Cutting it in three means solving a cubic, and for sixty degrees that cubic has no rational root — but plenty of angles do trisect, and which ones is a question with a countable answer.

Computation

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

Computation

The cube that will not double

Doubling a cube needs an edge in the ratio of the cube root of two. That number satisfies an equation of degree three, three does not divide any power of two, and the oldest open problem in geometry closes in a line.

Algebra

The group that will not come apart

Solving an equation by radicals means building a tower of roots, and a tower of roots corresponds to a chain of subgroups with abelian steps. For the general equation of degree five that chain would have to descend through a group of sixty elements with no normal subgroup in it — so there is no formula, and the obstruction is a finite object that can be written out.

Algebra

The integers a field contains

Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.

Algebra

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

Geometry

Three trisectors and a triangle nobody expected

Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.

Algebra

Units that form a lattice

In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.

Computation

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

Computation

Which polygons can be drawn

Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.

The whole library · What the figures prove