Fold and cut — the triangle
Fold it
The sheet is on the printed page, at the size it says.
Printing this page gives the page, and adds one more: the pattern alone, at 150 mm across, on a sheet of its own. Cut a square that size, transfer or trace the lines, and fold. The size is repeated in the corner of the sheet, because a printer set to fit to page rescales silently and there has to be some way to notice.
Mountain and valley are distinguished by dash as well as by colour, so the sheet survives the monochrome laser printer it will almost certainly come out of.
What it is
| Provenance | Generated here, from the straight skeleton |
|---|---|
| Creases | 2 mountain, 4 valley |
| Interior vertices | 1, every one checked |
| Panels | not read — see below |
| Folding to do | about 258 mm of crease at this size |
| Printed sheet | 150 mm across |
What was checked
Four theorems at every interior vertex, and the faces two ways.
- Developability — the sectors around each of the 1 interior vertices sum to a full turn, so the sheet was flat before it was creased.
- Kawasaki — alternating sectors sum to a straight angle at each of them.
- Maekawa — mountains and valleys differ by exactly two.
- Big-little-big — no strictly smallest sector is flanked by two creases of the same letter.
- The faces — not read, and not exported. This pattern's graph is in 2 pieces: the outline is where the scissors go rather than a crease, and it does not reach the edge of the sheet. Faces are read out of adjacency, and two components that never touch carry no adjacency saying which contains which — so the export carries edges only rather than faces that would be wrong.
None of this decides whether the whole sheet folds flat, which is NP-hard in general. Every local condition holds. That is a different and weaker statement, and it is the one being made.
Take it away
The field's own interchange format, so the pattern is reusable outside this site.
fold-and-cut-triangle.fold — 11 vertices, 16 edges, 1696 bytes. It opens in ORIPA, Rabbit Ear and the rest of the FOLD ecosystem.
The export is short because this repository never converts anything: FOLD's
vertices_coords, edges_vertices and edges_assignment
have been the in-memory representation of a pattern here since the site's first phase. What
the file adds is the metadata that makes it openable, and the faces where they can be read.
Coordinates are in sheet widths, and the file says what one unit measures on paper.
What is argued with it
Essays that call fold-and-cut — read off the figure index rather than listed by hand.
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.
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.
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.
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.
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.
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.