Matpic
mathematics, in pictures
Ladder

Constructible numbers — the ladder

6 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

    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.

    rung 1 · computation
  2. 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 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.

    rung 2 · computation
  3. Every rational number that could be a root of x³ − 2. A table of the candidate rational roots allowed by the rational root theorem, with the polynomial's exact value at each.

    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.

    rung 3 · computation
  4. Which angles with a rational cosine can be cut in three. A dial of angles marked trisectable or not, beside the cubic whose rational roots decided each one.

    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.

    rung 4 · computation
  5. Which regular polygons a compass and straightedge can draw, up to 100. A grid of the integers with the constructible ones filled in, each verdict computed two independent ways.

    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.

    rung 5 · computation
  6. Looking for a polynomial with π as a root. A table of the closest an integer polynomial of each degree comes to vanishing at the number, over a bounded search.

    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.

    rung 6 · computation

All ladders