A disc unrolled into a triangle
unroll 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
A disc unrolled into a triangle
More rings, straighter edge
A circle trapped between two 12-sided polygons
A hemisphere and a cylinder with a cone taken out, sliced at one height
Where slicing gives the area, and where it does not
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.
- and the circumscribed value is 6 tan(π/6) ×7
- and the inscribed value is 6 sin(π/6) ×7
- the 6-gons bracket pi ×7
- and along every ray the inner chord is exactly half the outer one while the areas are a quarter and three quarters ×1
- and by ninety-six sides the bracket sits inside Archimedes' own two fractions ×1
- and the whole ball is two of those ×1
- at every height the dome's disc and the ring have the same area ×1
- between three and seven doublings are tabulated ×1
- each doubling tightens the bracket from both sides ×1
- so the hemisphere's volume is the cylinder's less the cone's ×1
- the areas are integrated at between four hundred and twenty thousand slices ×1
- the drawn polygon has between three and twenty-four sides ×1
- the rings account for the whole disc ×1
- the shear leans by between a tenth and one and a fifth of the height ×1
- the sheared stack has the same area ×1
- the slice is taken strictly between the base and the top ×1
- the triangle of base 2πr and height r has the disc's area ×1
- the two regions together are the unit square ×1
- the two solids are compared at between a hundred and four thousand heights ×1
- the view is one the family draws ×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.
A circle unrolled into a triangle
Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.
GeometryPinned between two sequences
The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.
GeometryThe most area a fence can hold
One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.
GeometryThe slice that has to match
The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.