Cutting a Möbius band down the middle
mobius-cut 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
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 one lap leaves one loop while two laps leave two ×1
- the cut is inside the band ×1
- the scissors take one lap down the middle and two off-centre ×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.
A disc sewn to a Möbius band
The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.
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.
TopologyWhich 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.