Generator

A point, a ray, and 8 crossings

A generator in the topology library, called 39 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.

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

Four ways to say which pixels touch

The theorem holds only when the two neighbourhoods differ

A curve that divides the plane into three

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 fill spreading through corners escapes through a diagonal step of the outline ×1
a fill spreading through edges stays inside and covers about the enclosed area ×1
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
an acute triangle carries three squares, a right-angled two, an obtuse one ×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
at every proportion sampled the octagon carries a rectangle ×1
away from squares the count of rectangles is even at all but isolated proportions ×1
away from the instants of contact the count changes by two ×1
between 2 and 36 rays are drawn ×1
consecutive squares in the visiting order are neighbours ×1
each found rectangle has the requested proportions ×1
each square opens into two rectangles as the proportion leaves one ×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 polygon in the sample carries an odd number of squares ×1
every simple closed curve of eight-neighbours leaves exactly two pieces of four-neighbours ×1
every simple closed curve of four-neighbours, larger than a 2 × 2 block, leaves exactly two pieces of eight-neighbours ×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 simple closed curve of eight-neighbours separates anything when the rest also uses eight ×1
no two of the kept squares overlap ×1
on the honeycomb every simple closed curve leaves exactly two pieces ×1
read with eight neighbours the same pixels are connected but not a simple curve ×1
some sampled point is inside the curve ×1
some simple closed curve of four-neighbours leaves three pieces of four-neighbours ×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 count is odd along the path except at isolated instants ×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 outline is one piece when corners count ×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 octagon carries rectangles of each of these proportions ×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 same curve leaves three pieces of four-neighbours and two of eight ×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
this octagon carries exactly five inscribed squares ×1
two pitches, both inside the range the family draws ×1
under the mixed rule, parity along each row marks exactly the piece the curve cuts off ×1
with eight neighbours for both, the inside leaks out through the corners ×1
with eight neighbours for the curve and four for the rest, the diamond is one piece and leaves two ×1
with four neighbours for both, the diamond falls into twelve single pixels and still leaves two pieces ×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 of pixels needs two kinds of touching

On a screen a closed curve is a ring of pixels, and whether it has an inside depends on which pixels count as touching. Let diagonal pixels touch, and the inside leaks out through the corners; forbid it, and a curve can fence in a region while being twelve separate dots. Of the 1,048,575 sets of pixels in a small window, every simple closed curve obeys the Jordan curve theorem exactly when the curve and the rest of the plane are given different rules.

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

An odd number of squares on every polygon

Does every closed curve that does not cross itself pass through the four corners of some square? Toeplitz asked in 1911, and for polygons and smooth curves the answer is yes, because the squares can be counted and the count is odd. For rectangles of a fixed shape the count is even, which is why they needed a different proof — and for an arbitrary continuous curve the square is still not proved to exist.

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