Generator

A Möbius band

A generator in the topology library, called 34 times across 10 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.

mobius 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 Möbius band. A strip joined end to end after a half twist, so it has one side and one edge.

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 ledger's bound for dimension 1 is one less than the next power of two ×32
at dimension 2 the lower bound meets Whitney's ×5
after half a turn the first arrow points the opposite way ×1
and between 24 and 140 the long way ×1
and so does the second, which is exactly what the double flip at the join demands ×1
and the band's most economical section is zero exactly once ×1
between four and nine stations ×1
between none and sixteen cross-sections are outlined ×1
between one and four sections are drawn ×1
between one and three half-twists ×1
between three and nine frames are drawn along the band ×1
both rows the figure is about are drawn ×1
carrying the surface round once brings it back on the other face ×1
each folded band has length k tan(π/2k), above π/2 ×1
each section meets itself at the join ×1
each section meets itself at the join with its sign changed ×1
every section of the cylinder crosses the centre line an even number of times ×1
every section of the Möbius band crosses the centre line an odd number of times ×1
going once round the long way returns to the point with the cross-section reversed ×1
going once round the tube returns to the same point ×1
on its centre circle the curvature is −1/4 ×1
the band is drawn at a whole number of half-twists ×1
the cylinder has a section that is never zero ×1
the edge runs once round the triangle ×1
the folded strip closes with a flip ×1
the folded strip lies in an equilateral triangle of height 1 ×1
the frame comes back the same size — only its sense can change ×1
the frame returns with a reversed sign exactly when the band is one-sided ×1
the gluing arrow matches the twist the figure is about ×1
the inverse ledger is the row of Pascal's triangle the next power of two predicts ×1
the lengths fall as the folds increase ×1
the mesh is between 16 and 96 round the tube ×1
the section's sign at the join is fixed by the parity of the half-twists ×1
the self-intersection circle is on or off ×1
the single edge has length twice the strip's ×1
the tangent and normal ledgers multiply to the room's, which is 1 ×1
the textbook band is curved like a saddle at every point sampled ×1
the tube's distance from the axis leaves the figure-eight room to close ×1
the turning frame never degenerates and never changes its sense ×1
the view is one the family draws ×1
three and nine folds join with a flip, six folds join straight ×1
three folds at sixty degrees close a strip of length √3 with a flip ×1
u = 0 and u = π are two different places on the surface and one place in space ×1
which is not the same point unless it is one the reversal fixes ×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 loop that cannot be pulled tight

A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.

Topology

Every section of the band must vanish

Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.

Topology

Every surface is a sphere with handles

Take a square and say which edges are to be glued to which, and which way round. Four such rules give four different surfaces — and two numbers computed from the rule, without ever building the surface, say which one.

Topology

How short a paper Möbius band can be

A strip of paper ten times as long as it is wide makes a Möbius band easily. A square does not: paper bends but does not stretch, and a half-twist has to be paid for in length. Fold a strip of length √3 three times at sixty degrees and its ends meet turned over — a Möbius band, flat, in the shape of a triangle. Five folds manage it with 1.6246, and more folds approach π/2, but only by letting the paper pass through itself. Richard Schwartz proved in 2023 that √3 is the true limit for a band that does not.

Topology

The surface with one side, and what happens when it is cut

A strip of paper, half a twist, and a join. The result has one side and one edge, and cutting it down the middle does not produce two of anything.

Topology

Orientation is a sign

Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.

Topology

The bottle that needs a fourth dimension

Take the Möbius band's rectangle and glue the second pair of edges too. The result is closed, one-sided, and cannot be built in three dimensions without passing through itself — which is a fact about the room rather than about the surface.

Topology

The room a projective space needs, read off Pascal's triangle

The projective plane cannot sit in three-dimensional space without crossing itself, and the reason can be written as arithmetic: a polynomial that records how a shape twists, which a room must cancel. For the n-dimensional projective space that polynomial is a row of Pascal's triangle read mod 2, its inverse is another row, and the inverse's last term says how many extra dimensions the room must have — exactly enough, at every power of two.

Topology

The same loop, unrolled

Spread a circle out into a line spiralling above it, and a loop that closes downstairs becomes a path that does not — so a question about which loops can be shrunk becomes a question about where a path ends, which is easy.

Topology

Two sheets over a one-sided surface

Above every one-sided surface sits a two-sided one, exactly twice as large, and the map between them forgets which of the two senses of turning a point was carrying. Building it turns a question about sides into a question about covers.

The whole library · What the figures prove