One straight cut: the triangle
fold-and-cut 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
shape: "triangle"
shape: "ell"
shape: "star"
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.
- developability holds at every interior vertex — its sectors close to 360° — 1 checked ×3
- every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×3
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 1 checked ×3
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 1 checked ×3
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 1 checked ×3
- every node of the straight skeleton sits the same distance from each edge that formed it, to 0e+0 — which is why one fold can carry several edges of the square onto the cut line at once ×1
- every node of the straight skeleton sits the same distance from each edge that formed it, to 2e-16 — which is why one fold can carry several edges of the house onto the cut line at once ×1
- every node of the straight skeleton sits the same distance from each edge that formed it, to 2e-16 — which is why one fold can carry several edges of the star onto the cut line at once ×1
- every node of the straight skeleton sits the same distance from each edge that formed it, to 2e-16 — which is why one fold can carry several edges of the triangle onto the cut line at once ×1
- every node of the straight skeleton sits the same distance from each edge that formed it, to 6e-17 — which is why one fold can carry several edges of the ell onto the cut line at once ×1
- every node of the straight skeleton sits the same distance from each edge that formed it, to 6e-17 — which is why one fold can carry several edges of the pentagon onto the cut line at once ×1
- every node of the straight skeleton sits the same distance from each edge that formed it, to 6e-17 — which is why one fold can carry several edges of the rectangle onto the cut line at once ×1
- the Fold and cut — ell is put past all four theorems before it is drawn ×1
- the Fold and cut — house is put past all four theorems before it is drawn ×1
- the Fold and cut — pentagon is put past all four theorems before it is drawn ×1
- the Fold and cut — rectangle is put past all four theorems before it is drawn ×1
- the Fold and cut — square is put past all four theorems before it is drawn ×1
- the Fold and cut — star is put past all four theorems before it is drawn ×1
- the Fold and cut — triangle is put past all four theorems before it is drawn ×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 tree cannot argue
A molecule fills a polygon with creases taken from its straight skeleton, and a straight skeleton is a tree. So a molecule's panels have almost no closed chains for its letters to contradict themselves round — one to three, against thirty-six on the smallest tessellation patch. Two hundred and eighty independent letterings across seven outlines, including an L and a five-pointed star, and not one of them disagrees with itself.
One cut for a star
The fold-and-cut construction here could reach a triangle, a pentagon and a house, and refused everything that turned back on itself, because shrinking an outline with a reflex corner needs an event the shrink did not implement. With split events it reaches a five-pointed star — ten creases through one point, four hundred and twenty letterings that fold, and every edge of the outline landing on one line to a part in 10^16.
One straight cut
Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.
The corner that splits the shrink
The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.
The oldest book cuts the paper
The Hiden Senbazuru Orikata of 1797 is the earliest surviving book of recreational paper folding, and its famous connected cranes are made by slitting one sheet into a grid. The founding rule of the modern subject is younger than the tradition it claims to describe.
The star that was cut before it was proved
Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.
The symmetry the letters cannot keep
Every pattern in this subject is drawn symmetric and the symmetry is always quoted of the drawing. A folded object is a drawing and a lettering together, so a symmetry survives only if the letters keep it — and the preliminary base loses every rotation while the square twist, drawn with the same eight, loses the other half.
The vertex the list does not have
Every condition this collection checks is asked at a vertex of a crease pattern, and a crease pattern is handed to the checker as a list of points and segments. A reader is handed ink. Read the same patterns the second way and eight printed sheets gain nothing at all — while four tessellation patches gain 12, 18, 12 and 5 vertices that nobody wrote down, every one of them a place where two creases were drawn across each other.
What universality costs
The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.
Every generator · The flat-folding field · The patterns a reader can fold