The same parity, with nowhere to put it
hole-colouring is one function. Everything below came out of it during this
build, at arguments 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 when the generator changes, this
page changes with it.
At its defaults
count: 3
view: "chi"
view: "pair", count: 3
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- a loop of paper with 3 creases from the hole to the rim has no interior vertices at all, so every vertex theorem in the subject holds on it vacuously ×3
- a ring of paper with 3, 5, 7 creases across it has no two-colouring and places its panels 1.75 to 1.75 sheet widths apart; cut open, every one of them two-colours and places exactly ×1
- a ring of paper with 3, 5, 7, 9, 11 creases across it has no two-colouring and places its panels 1.75 to 1.75 sheet widths apart; cut open, every one of them two-colours and places exactly ×1
- a sheet gains a parity condition for every loop it has that cannot be shrunk to a point, and cutting a closed hole adds one where cutting inward from the rim adds none ×1
- a sheet with two holes has two loops that cannot be shrunk, and each of them reads its own parity — satisfying one says nothing about the other ×1
- a two-holed sheet folds flat only when both loops cross an even number of creases, and cases satisfying exactly one of the two are refused ×1
- every closed path drawn round the hole crosses the same number of creases — 12 paths of three different shapes at four radii, all 3 ×1
- its panels refuse two colours, and the composition of reflections round the same cycle does not close — two computations sharing no code ×1
- its panels take two colours, and the composition of reflections round the same cycle closes — two computations sharing no code ×1
- the cut runs from the hole to the rim and crosses no crease, so every crease, letter and angle is the same on both sides of the comparison ×1
- the panel colouring and the composition of reflections agree about every two-holed sheet tried, on two computations sharing no code ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
A contradiction is even
A crease pattern's letters can demand a circle of panels each of which lies below the next, which is a proof that the sheet has no folded state. Every such circle found here — one thousand one hundred and forty-nine of them, across every family of patterns this collection draws — has an even number of panels in it, and none has four. Both facts are theorems rather than observations, and they come from opposite ends of the subject.
A cut is surgery
Two cuts that look identical on the paper do completely different things to the sheet. A slit run inward from the rim changes nothing at all; a closed cut in the middle removes a disc and leaves a sheet carrying a condition it did not have before. What separates them is not the length of the cut or how much paper it removes.
A cut that reaches the edge
A ring of paper with three creases running from its hole to its rim satisfies every condition the subject has — vacuously, because it has no interior vertex at all — and cannot be folded: its panels take no two colours and the two routes to one of them end up 1.75 sheet widths apart. One cut from the hole to the edge, crossing no crease and changing no letter, and it folds exactly. The cut removes an adjacency, which is the one thing neither a fold nor an edge can do.
A cut that removes no paper
Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.
A short reason to say no
When a folding question comes back yes it brings an object anybody can check. When it comes back no it usually brings nothing but the assurance that a search looked everywhere. At one vertex that is false: a refusal comes with a witness one or two steps long, out of a search space of a hundred and twelve, and the witness is a vertex the crease pattern does not contain.
A theorem with an unstated hypothesis
Maekawa's and Kawasaki's conditions are quoted everywhere without saying which sheet they are about, and they do not need to be — they are conditions at a point and every point is the same. The two-colouring is quoted the same way and it is not a condition at a point, and the omission there is not harmless.
Decided before the design
A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.
Even is not enough
Every vertex theorem in the subject is a statement about one point, and the two-colouring of the panels looks like the exception. It is not — on a square of paper it is a parity at each vertex and nothing more. Cut a hole and the two come apart: a loop of paper with three creases has no interior vertices at all, satisfies every theorem there is, and cannot be folded flat.
How rare a band that folds is
Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.
Parity is not enough
A Möbius band needs an odd number of creases round it. Give it three, square across the strip, and it does not fold — nor does five, nor seven, nor any odd number at all. The counting argument is necessary and it is not close to sufficient, and the thing it cannot see is which way the creases point.
The cut that changes nothing
A slit goes right through the material and leaves the sheet exactly the object it was. A closed cut removes almost no paper and produces a different sheet with a condition it did not have. Kirigami is made almost entirely of the first kind, which is why every result about it survives the distinction untouched.
The seam carries a sign
A loop of paper folds flat when it has an even number of creases round it. A Möbius band folds flat when it has an odd number. The drawing is the same in both cases, the creases are the same creases, and what changed is a factor of minus one contributed by the sheet rather than by anything drawn on it.
Two holes are two conditions
One hole in a sheet of paper gives one loop that cannot be shrunk and one parity to satisfy. Two holes give two, and they are independent: an arrangement of creases can satisfy the condition round one hole and fail the condition round the other, and the sheet refuses on the strength of the one it failed.
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