construct
construct 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: "root"
show: "trisect"
show: "polygons"
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
- a right angle can be trisected ×1
- a root found by the theorem really is a root of the cubic ×1
- a straight angle can be trisected ×1
- a zero angle can be trisected ×1
- and it sits over the midpoint ×1
- cos(θ/3) is a root of the triple-angle cubic ×1
- divisors are taken of a positive whole number ×1
- every point of the first round is where two drawn objects cross ×1
- Gauss's criterion and the degree test agree about every polygon ×1
- nineteen of the polygons up to sixty can be drawn ×1
- one round puts two points off the line ×1
- sixty degrees cannot be trisected ×1
- the apex sees the diameter at a right angle ×1
- the denominator the cosines are taken over is a whole number between 2 and 16 ×1
- the largest polygon tested is a whole number between 12 and 300 ×1
- the new height is √3/2 ×1
- the nine-gon is not — 3 is a Fermat prime but 9 repeats it ×1
- the number being factorised is a whole number between 1 and 1000000 ×1
- the number of columns in the grid is a whole number between 6 and 20 ×1
- the number of construction rounds is a whole number between 1 and 2 ×1
- the number whose square root is constructed is between 0 and 12 ×1
- the perpendicular at the join has height √n ×1
- the second round is shown or not ×1
- the seven-gon is not ×1
- the seventeen-gon is constructible ×1
- twenty-four of the polygons up to a hundred can be drawn ×1
- two points and one round of drawing give four more ×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.