Generator

A point, a ray, and 8 crossings

A generator in the topology library, called 28 times across 5 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.

parity 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 point, a ray, and 8 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Alexander's horned sphere at stage 3

Every ear carried to a slice of a disc

Stage 3 of a curve that has area

A closed curve with a tangent nowhere

The same area, and no room in it

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.

none of the 594 pairs of non-neighbouring edges meet ×5
a pair of horns at stage 1 is clasped, of linking number ±1 ×4
stage 3 has 2^3 − 1 clasped pairs ×4
ear clipping: a polygon of 20 corners is cut into 18 triangles ×2
the image triangulation: a polygon of 20 corners is cut into 18 triangles ×2
the source triangulation: a polygon of 20 corners is cut into 18 triangles ×2
a line with a point removed falls into two pieces ×1
a plane with a line removed falls into two pieces ×1
a plane with a point removed stays in one piece ×1
a triangulation of a polygon has one diagonal fewer than it has triangles ×1
all 16 rays agree on the parity ×1
and a ray from the centre crosses it an even number of times ×1
and both insides keep a substantial part of the frame ×1
and is not linked with that horn's partner ×1
and the clasp shows as at least two crossings of the projection ×1
and the steepest chord anywhere grows geometrically as the scale shrinks ×1
and they disagree on the count, which is the point ×1
at every angle tried the chords steepen by more than fifteenfold over the scales measured ×1
between 2 and 36 rays are drawn ×1
consecutive squares in the visiting order are neighbours ×1
each square's side is at most 2⁻ⁿ, so the diameters go to zero ×1
ear clipping: no two triangles overlap ×1
ear clipping: the triangles account for exactly the polygon's own area ×1
every cell of the grid is visited exactly once ×1
every direction gives the same parity ×1
every direction of ray gives the same verdict at this point ×1
every simple polygon of four or more corners has an ear to clip ×1
halving the pitch more than halves the largest disc that fits ×1
horns from different clasps are not linked with each other ×1
no two of the kept squares overlap ×1
some sampled point is inside the curve ×1
space with a line removed stays in one piece ×1
the arms of the spiral are between 22 and 70 units apart ×1
the clipping terminates ×1
the counts themselves differ, which is why the parity is the claim ×1
the curve goes twice round its own centre ×1
the disc drawn touches no edge of the curve ×1
the drawn ray crosses the surface more than once ×1
the five-pointed loop crosses itself five times ×1
the horns shrink geometrically, so their tips converge ×1
the image triangulation: no two triangles overlap ×1
the image triangulation: the triangles account for exactly the polygon's own area ×1
the infinite product is bounded away from zero ×1
the kept area is the product of the stages' own factors ×1
the loop drawn goes once round one horn ×1
the magnification of the second panel is between 3 and 27 ×1
the marked point is inside the frame, given as fractions of it ×1
the number of stages drawn is between 1 and 4 ×1
the number of stages of horns is between 1 and 4 ×1
the number of terms summed is between 2 and 6 ×1
the outside point is clear of the tube ×1
the plane comes apart into exactly two pieces, no more and no fewer ×1
the point is put inside or outside ×1
the radius stays positive, so no two angles share a point ×1
the resolution of the flood fill is between 24 and 90 ×1
the sample grid is between 8 and 40 columns ×1
the sampled inside agrees with the polygon's own area to within six per cent ×1
the second corridor is the narrower one ×1
the source triangulation: no two triangles overlap ×1
the source triangulation: the triangles account for exactly the polygon's own area ×1
the spiral turns between 1 and 6 times ×1
the terms roughen faster than they shrink ×1
the tube is closed up out of two triangles per patch ×1
the two triangles either side of a diagonal send a point on it to the same place ×1
the view is one the family draws ×1
two pitches, both inside the range 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.

Topology

A ball whose outside is not one

Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.

Topology

A curve that has area

The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.

Topology

Every loop is a circle in disguise

Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

Topology

Two pieces, in every dimension

A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.

Topology

Which side of the line is inside

A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.

The whole library · What the figures prove