Origami & Folding
what one sheet can do
Generator

The period cell of the Yoshimura

A generator in the tessellations library, called 66 times across 15 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.

family-glued 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

The period cell of the Yoshimurathe Yoshimura drawn over the plane, with one period rectangle marked on it and a ring of its neighbours around it. The rectangle's edges are placed to miss every vertex, so identifying opposite edges can neither make nor destroy an interior vertex — there are 2 of them either way. one column wide and two courses high, because the courses alternate their offset.the period cell of the Yoshimuraone period, with its neighbours round it2 interior vertices in the cell12 crease pieces drawnperiod 0.167 × 0.289one column wide and two courses high, because the courses alternate their offsetthe cell is a rectangle of ordinary paper until somebody says its edges are one edge

view: "patch", family: "grid"

The period cell of the gridthe grid drawn over the plane, with one period rectangle marked on it and a ring of its neighbours around it. The rectangle's edges are placed to miss every vertex, so identifying opposite edges can neither make nor destroy an interior vertex — there are 1 of them either way. one square, because a grid repeats at every line.the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge

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

The two ways a gluing failsFor each drawing, size and direction, whether the gluing closes and — where it does not — which of the two failures it is. A gluing can bring the paper back the other way up, which is a parity and kills the two-colouring; or it can bring it back turned through an angle, which means the drawing's period is not the folded state's. No sheet here does both.the two ways a gluing failsthe grid ×1 xflipcomes back turned over — 1 creases crossedthe grid ×1 yflipcomes back turned over — 1 creases crossedthe grid ×2 x—closesthe grid ×2 y—closesthe grid ×3 xflipcomes back turned over — 3 creases crossedthe grid ×3 yflipcomes back turned over — 3 creases crossedone is a parity and the other is an angle, and one number was reporting both

view: "period", family: "grid", max: 5

How many columns the grid takes to repeat when it is foldedFor each number of drawn periods, the turn the fold applies between one cell and the next, and whether the resulting glued sheet keeps the paper the same way up and relates its cells by a slide. the grid has a drawn period of one and a folded period of 2.the folded period of the griddrawn periods across the top12345the turnsame way upslides0°noyes0°yesyes0°noyes0°yesyes0°noyesa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

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 glued

Box pleating is the designer's grid: every crease on a line, every angle a right angle or forty-five degrees, and a whole design method built on the convenience of it. Roll the grid into a tube and half the column counts stop folding, on a pattern whose whole selling point is that it always works.

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 map with no edges

Counting the ways a rectangular map folds is the oldest open problem in the subject, and every version of it assumes the map has an edge. Join the map's opposite edges and the question changes shape: half the sizes have no folded state at all, and the ones that do have no bottom layer to count from.

A mechanism that closes on itself

A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.

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.

Nothing grown has a seam

A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.

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.

The arc that arrived twice

Which of two panels a crease calls its near one is decided by the order a face walk happened to number them, and the mirrored record is the same relation. Except on one sheet, where it is not — and that sheet turned out to be the one whose folded state comes back the other way up, which is how a duplicate in a graph became a diagnosis.

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 period nobody measured

Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.

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 symmetry a gluing adds

A patch of a tessellation has whatever symmetry its outline allows — a few reflections, a rotation or two. Glue its edges and it acquires translations, and a lettering of the glued sheet has to be invariant under them. That is a much stronger requirement than a lettering of the patch, and it is why one answer covers every patch at once.

The tube a map makes

Join one pair of a map's edges and the result is a tube — a real object, foldable in the hand, and neither the strip's problem nor the torus's. It has one loop that cannot be shrunk instead of two, it keeps its bottom layer because it keeps half its rim, and half its sizes are refused by a parity the flat map does not have.

The turn a column costs

The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.

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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