Origami & Folding
what one sheet can do
Generator

One rectangle, glued four ways

A generator in the flat-folding library, called 85 times across 20 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

rim-ladder 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

One rectangle, glued four waysThe same rectangle of paper with the same creases on it, four times: cut out of the plane in the ordinary way, with its left and right edges declared to be one edge, with its top and bottom edges declared to be one edge, and with both. Matching arrowheads mark the pairs. Nothing in the crease pattern distinguishes the four, and each of them is a different sheet of paper.one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

view: "parities", families: [grid, miura], cells: [1, 2, 3]

Two ways to ask whether a gluing turns the paper overFor each glued sheet, the number of creases a loop that cannot be shrunk crosses on the flat drawing, and beside it what the folded motions say about the same gluing. The first is a count and the second is a comparison of six numbers; they share no code and they agree everywhere.creases crossed by a loop, and what the fold says about itthe grid ×1, across11 creases, always odd · turns the paper overthe grid ×1, along11 creases, always odd · turns the paper overthe grid ×2, across22 creases, always even · keeps the sidethe grid ×2, along22 creases, always even · keeps the sidethe grid ×3, across33 creases, always odd · turns the paper overthe grid ×3, along33 creases, always odd · turns the paper overthe Miura ×1, across11 creases, always odd · turns the paper overthe Miura ×1, along42–4 creases, always even · keeps the sidethe Miura ×2, across22 creases, always even · keeps the sidethe Miura ×2, along44–8 creases, always even · keeps the sidethe Miura ×3, across33 creases, always odd · turns the paper overthe Miura ×3, along126–12 creases, always even · keeps the sidean odd count and a folded state that comes back the other way up are the same fact

view: "ladder", family: "miura"

the Miura on four sheetsFour counts for one rectangle of the Miura pattern, on each of the four sheets its edges can be glued into. The interior vertex count does not move, because the cell's edges are placed to miss every vertex; the free letters and the panels fall as the rim goes; and Euler's number is 1 on the cut cell and 0 on the other three.the Miura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters22182016panels1510128V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

view: "parities"

Two ways to ask whether a gluing turns the paper overFor each glued sheet, the number of creases a loop that cannot be shrunk crosses on the flat drawing, and beside it what the folded motions say about the same gluing. The first is a count and the second is a comparison of six numbers; they share no code and they agree everywhere.creases crossed by a loop, and what the fold says about itthe grid ×1, across11 creases, always odd · turns the paper overthe grid ×1, along11 creases, always odd · turns the paper overthe grid ×2, across22 creases, always even · keeps the sidethe grid ×2, along22 creases, always even · keeps the sidethe Miura ×1, across11 creases, always odd · turns the paper overthe Miura ×1, along42–4 creases, always even · keeps the sidethe Miura ×2, across22 creases, always even · keeps the sidethe Miura ×2, along44–8 creases, always even · keeps the sidethe Yoshimura ×1, across22 creases, always even · keeps the sidethe Yoshimura ×1, along44 creases, always even · keeps the sidethe Yoshimura ×2, across44 creases, always even · keeps the sidethe Yoshimura ×2, along88 creases, always even · keeps the sidean odd count and a folded state that comes back the other way up are the same fact

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.

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 grid that will not close

Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.

A metamaterial with no edge

A folded metamaterial's properties are quoted per unit cell, because a material is supposed to be the same everywhere and a cell is supposed to stand for the whole of it. Every cell this collection has measured has been cut out of a patch, with a rim round it — and a rim is the one place a repeating material is not like itself.

A proof in no nodes at all

A parity refuses a sheet before any search begins. It costs one addition, it is certain, and it says nothing about why — while a search that exhausts on the same sheet costs thousands of nodes and produces a proof of the same fact. Two proofs of one thing, and the cheap one is available only where somebody has noticed the invariant.

A sheet with two edges

Design in this subject starts from a square, and the square's four edges are where every flap ends and every construction begins. A cylinder has two edges instead of four, and what a designer loses turns out to be measurable in the same letters a search counts — and what they gain is that the sheet is what gets manufactured.

A tessellation on a cylinder

A twist tessellation has been drawn here as a patch and as a torus, and never as anything in between. Gluing one pair of a cell's edges gives the family its first sheet with exactly two edges — the shape every folded tube actually has, and the only object in the collection that can say whether the rim's cost is linear in how much rim there is.

Euler counts the gluing

Vertices minus creases plus panels comes to one on a rectangle of paper and nought on any gluing of it. That is the cheapest check that an identification did what it says, it costs three counts already being made, and it is what found a crease running exactly through the corner of a cell — a case the corner search could not see and no other check would have noticed.

Half a rim

A rectangle of tessellation cut out of the plane has four edges; glued into a torus it has none. Gluing one pair and leaving the other gives the middle of the scale — the same drawing, the same vertices, the same conditions asked of them, and exactly half the rim. What the rim costs turns out to be measurable per edge rather than only at the ends.

Half the slack

Gluing one pair of a cell's edges removes half the free letters and costs almost nothing. Gluing the second pair removes the other half and costs three orders of magnitude. The letters go linearly and the search does not, and the reason is that the last free letter is worth more than all the others.

No format has a gluing

A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.

One node per panel, with the rim gone

A rectangle of repeating pattern cut out of the plane costs exactly one node of search per panel, on every family and at every size. Take the rim away and the total falls and the cost per panel rises, because the letters that were removed were the ones that could not be wrong.

One population, four sheets

A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.

The corrugation that closes on itself

A Miura cell crosses one crease per period in one direction and four in the other. So one of its two directions has a parity condition that half the sizes fail, and the other direction cannot fail at any size — the same sheet, the same drawing, and two gluings that behave completely differently.

The drawing does not say what is glued

One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.

The rim adds up

What one glued pair of a cell's edges saves in free letters is what the other pair saves, and gluing both saves the sum. That is a rate rather than an observation, it is the form of the claim two objects could never support, and it is what makes 'the rim costs four letters a cell' a statement about tessellations rather than about one drawing.

The seam that is not a symmetry

Gluing a cell's edges looks like a symmetry of the drawing and is not. It is an instruction about which points of the paper are the same point, the drawing has to agree with it along the whole of a glued edge, and a rectangle that is not a period of the pattern does not glue at all — which turns out to be the only real restriction on which cylinders exist.

The sixth thing that is not true

Five idealisations underlie every theorem here and each has had an essay: no thickness, no stretch, creases that are lines, perfect memory, no grain. There is a sixth, it is more basic than any of them, and it is the one nobody has ever thought to name — the paper is a disc.

The tube that gets built

Every folded structure that leaves a laboratory is a sheet joined to itself — a boom, a stent, a bellows, an airbag, a packed antenna. The mathematics has been done on flat rectangles for the whole history of the subject, and the object is a cylinder, which is a different sheet with different counts and a condition the rectangle does not have.

Thickness round a closed loop

Real panels have thickness, and every technique for accommodating it works by shifting a hinge off the ideal crease by a small amount. On a flat sheet the shifts accumulate outward and end at the edge. On a closed sheet they accumulate round a loop and have to come back to where they started, which is a condition none of the techniques was designed to satisfy.

Which of the seven survive

The seven axioms are the complete list of ways one fold can be specified by aligning marked things. Every one of them names points and lines on a sheet, three of them quietly assume that a line has two sides, and on a closed sheet a line need not — so the list is complete for a disc and shorter for anything else.

Which pair is glued

A cell's two cylinders have the same Euler number, the same amount of rim and the same name. On a symmetric drawing they have identical counts of letters, panels and vertices — and searching them costs twenty-four nodes one way and eighty-five the other. Half the rim is a description of the topology and not of the object.

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