Generator

The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3)

A generator in the computation library, called 74 times across 18 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, √5 and 1/3 are distances already; the golden ratio is not yet; ∛2 and π are not ×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 number missing from the line is missing ×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
between twenty and fifty-two primes are drawn ×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 field's points are symmetric about the line through the starting points and about their perpendicular bisector ×1
each new square root here is of a rational number ×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 coefficient of the class polynomial is a whole number to thirty places ×1
every degree divides the degree of the field, six ×1
every factor pattern of x⁷ − 7x + 3 is the cycle type of a collineation ×1
every factor pattern that occurs is the cycle type of an element of the group ×1
every measured share is within a percentage point of its share among the collineations ×1
every measured share is within a percentage point of its share in the group ×1
every pattern that occurs is a cycle type of the Galois group ×1
every point is rational, in ℚ(√3), or needs one more square root ×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
four to ten witnesses are listed ×1
it splits completely exactly when 2 and 3 are both squares ×1
j at the one reduced form is minus the cube ×1
modulo a prime leaving 3 on division by 8, 2 and −1 are non-squares and −2 is a square ×1
modulo every prime, x³ − x − 1 has as many roots as the class polynomial of −23 ×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 collineation has cycle type 1 + 2 + 4 ×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 census runs to a thousand, ten thousand or a hundred thousand ×1
the census runs to ten thousand or a hundred thousand ×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 computed counts are 6 and 203 ×1
the control polynomial really vanishes at its algebraic number ×1
the cube root really cubes to it ×1
the degree equals the number of reduced forms ×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 discriminant has three classes ×1
the discriminant has two or three classes ×1
the earlier powers are independent, so the solve is unique ×1
the element has the same cycle type on both sets of roots ×1
the element is k ↦ uk + b with u odd and b from 0 to 7 ×1
the exact value and the decimal one agree ×1
the expansion begins 1/q + 744 + 196,884 q + 21,493,760 q² ×1
the factor degrees add to the degree ×1
the factor degrees add up to the degree ×1
the field has one class ×1
the field is one the family knows ×1
the fifteen points on the line are the ones named ×1
the five cycle types account for all 168 collineations ×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 invertible modulo p ×1
the leading coefficient is not zero ×1
the list of patterns covers every cycle type of the group ×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 of roots modulo p matches the forms that take p ×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 shortfall is close to 196,884 e^(−π√d) ×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 value of j is a perfect cube ×1
the view is one the family draws ×1
two rounds give 6 and then 203 points, in floating point and exactly ×1
x³ − 3x + 1 splits exactly when p ≡ ±1 mod 9 ×1
x⁴ − 10x² + 1 is never irreducible modulo a prime ×1
x⁸ − 3 and x⁸ − 48 factor alike modulo every prime not dividing 6 ×1
x⁸ − 48 has as many roots modulo p as x⁸ − 3 ×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.

Algebra

The class number is a degree

The integers of ℚ(√−23) fall into three classes of ideals, and that 3 can be found by counting forms. It is also the degree of a single number: the modular function j, evaluated at the three forms' points, gives three values that are not whole but are the roots of one cubic with whole-number coefficients. When there is only one class, the value is a whole number outright — and that is why e^(π√163) misses being whole by less than a trillionth.

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

Two fields that every prime confuses

Reduce x⁸ − 3 and x⁸ − 48 modulo any prime other than 2 and 3 and factor them: the two lists of factor degrees are the same, at all 9,590 such primes below 100,000. The fields the two equations define are still different. The reason is a pair of ways for one group to permute eight roots, with the same cycle lengths element by element and different fixed points — the same coincidence, in degree seven, as the points and lines of the Fano plane.

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 build in two rounds

Start with two points and draw every line and circle they allow; mark every crossing; do it again. The first round gives six points, the second gives 203, and the third gives over 1.7 billion. Computed exactly, the 203 points show the tower of square roots growing in real time — eleven rational points, seventy-six that need √3, and the rest needing one of six new square roots.

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