degree
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.
At its defaults
show: "roots"
show: "search"
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.
- a rational is a whole numerator over a non-zero whole denominator ×1
- divisors are taken of a positive whole 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
- every coefficient is a whole number under 400 ×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
- no integer polynomial in the searched range vanishes at π ×1
- the closest miss is a genuine miss ×1
- the coefficient bound is a whole number between 2 and 8 ×1
- the control polynomial really vanishes at its algebraic number ×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 exact value and the decimal one agree ×1
- the largest degree searched is a whole number between 1 and 4 ×1
- the leading coefficient is not zero ×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 polynomial has degree between one and four ×1
- the polynomial is named for the caption ×1
- the product is the coefficient times the basis element it lands on ×1
- the same search does find a polynomial for an algebraic number ×1
- the searched number is a root of one of them ×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
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
ComputationThe 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.
ComputationThe 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.
ComputationThe 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.
ComputationWhat 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.
ComputationWhich 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.