Constructible number
Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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 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.
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.
The straightedge buys nothing
Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.
The mark that changes what is reachable
Two thousand years of failure to trisect an angle with compass and straightedge was failure at a stated set of operations. Scratch two marks on the straightedge and Archimedes trisects any angle in four steps — because the new operation solves a cubic, and the old ones could only ever solve quadratics.
One circle, and a straightedge
A straightedge alone cannot bisect a segment, so it cannot draw a parallel, so it can construct almost nothing. Draw one circle anywhere and mark its centre and everything a compass could ever have done becomes available — the circle is never needed again.
The compass that will not open
Fix the compass at one opening and never change it. That looks like a serious loss — a circle of a given radius through a given point is the compass's whole job — and it turns out to cost nothing at all, for reasons that are arithmetic rather than geometric.
Named alongside it
The objects these essays reach for when they reach for this one.
Degree of an extensionStraightedge and compassOperation setField extensionMinimal polynomialRational root theoremCircleConstructionIrreducible polynomialMidpointStraightedgeBasis