Origami & Folding
what one sheet can do
Concept

Torus — where it appears

A rectangle of paper with both pairs of opposite edges identified. It has no boundary at all, so it has no bottom layer and no free letters from a rim, and it is the closest a finite object comes to being a tessellation itself.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

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

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.

flat-folding · Boundary the Yoshimura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters36283224panels29202416V − 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

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.

flat-folding · Boundary 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

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.

flat-folding · Two-colourability the folded period of the Yoshimuradrawn periods across the top123456the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyesa 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

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.

tessellation · Shrinkage the grid, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices4444free letters1210108panels9664V − 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

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.

complexity · Typical instances 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

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.

complexity · Map folding the period cell of the Miuraone period, with its neighbours round it2 interior vertices in the cell7 crease pieces drawnperiod 1.000 × 2.000one column wide and two rows high, because the zigzag returns after twothe cell is a rectangle of ordinary paper until somebody says its edges are one edge

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.

tessellation · Mechanical metamaterials 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

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.

design · Box pleating

Named alongside it

The objects these essays reach for when they reach for this one.

GluingBoundaryPeriodicityPatchGridPanelParityUnit cellCrease assignmentThe folded periodInterior vertexBox pleating

All concepts