A Möbius band
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.
At its defaults
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 between 24 and 140 the long way ×1
- between none and sixteen cross-sections are outlined ×1
- between three and nine frames are drawn along the band ×1
- carrying the surface round once brings it back on the other face ×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
- the band is drawn at a whole number of half-twists ×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 mesh is between 16 and 96 round the tube ×1
- the self-intersection circle is on or off ×1
- the tube's distance from the axis leaves the figure-eight room to close ×1
- the view is one the family draws ×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
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
TopologyEvery 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.
TopologyThe 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.
TopologyOrientation 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.
TopologyThe 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.
TopologyThe 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.
TopologyTwo 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.