Generator

A disc unrolled into a triangle

A generator in the geometry library, called 33 times across 7 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

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. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

Bands of a sphere and of its cylinder, cut by the same planes

Why a band of a sphere has its cylinder's area: the radius shrinks as the slant grows

The surface of a stack of inscribed cones, closing on four times the great circle

Points chosen at random on a sphere: by even height, and by even latitude

The sphere unrolled from its cylinder: a map in which every area is true

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.

the recursion and the gamma function agree on the 0-ball ×21
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
a thin band well inside the sphere ×1
an even number of sides for each half-polygon ×1
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 in ten it reaches out past them ×1
and its surface is largest in seven ×1
and more with every doubling ×1
and the surface peaks exactly two dimensions later ×1
and the whole ball is two of those ×1
as a smooth function of the dimension, the volume peaks a little past five ×1
at every height the dome's disc and the ring have the same area ×1
between 200 and 4000 points ×1
between three and seven doublings are tabulated ×1
cos²θ is a root of the remaining factor ×1
dimensions up to between 10 and 30 ×1
each band is a pair of heights between −1 and 1, lower first ×1
each cell of the map has the area of the cell of the sphere it shows ×1
each doubling tightens the bracket from both sides ×1
each slice density integrates to one ×1
heights chosen evenly put the share the cap's area predicts in the cap ×1
in nine dimensions the middle ball just reaches the faces of the cube ×1
latitudes chosen evenly put the share of latitude in it instead ×1
one label for each band ×1
only even powers of cos θ occur ×1
so the hemisphere's volume is the cylinder's less the cone's ×1
square roots reach exactly the five ×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 even-dimensional volumes add up to e to the π ×1
the graticule is every 10, 15 or 30 degrees ×1
the inscribed stack of cones has less surface than the sphere ×1
the lune is the inner triangle less the outer one ×1
the lune on the leg has exactly half the triangle's area, which is 1/2 of 1 ×1
the lune's area equals the difference of two triangles ×1
the resolvent cubic has whole coefficients after scaling ×1
the rings account for the whole disc ×1
the sectors are equal ×1
the share within the outer tenth is one less nine tenths to the power n ×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 sphere's band has the area of the cylinder's band of the same height ×1
the stack's surface is 4π cos(π/2k), a closed form Archimedes could bound ×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 sectors are equal, so π cancels ×1
the two solids are compared at between a hundred and four thousand heights ×1
the unit ball is largest in five dimensions ×1
the view is one the family draws ×1
the whole sphere has four times the area of its great circle ×1
two to five dimensions ×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.

Geometry

A band of a sphere is a band of its cylinder

Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.

Geometry

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.

Geometry

Pinned 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.

Geometry

The ball that is largest in five dimensions

A disc of radius one has area π, a ball of radius one volume 4π/3, and in each further dimension the unit ball grows — until five, where its volume is 5.264, after which it shrinks towards nothing. The recursion that shows it is the ring dissection of the disc, done one dimension at a time, and the shrinking is not the ball getting small: it is almost all of a high-dimensional cube lying outside the ball, and almost all of the ball lying in a thin rind at its surface.

Geometry

The five crescents that can be squared

The circle cannot be squared, but some regions bounded by circular arcs can: Hippocrates of Chios found a crescent with exactly the area of a triangle around 440 BC. The trick is to make two multiples of π cancel, and it works for a crescent whose two arcs turn through angles in the ratio m to n exactly when an angle defined by a polynomial can be drawn with ruler and compass. That happens for five ratios and no others.

Geometry

The 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.

Geometry

The 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.

The whole library · What the figures prove