Origami & Folding
what one sheet can do
Concept

Cylinder — where it appears

A rectangle of paper with one pair of opposite edges identified, which is the shape every folded tube has. It carries one loop that cannot be shrunk, and therefore one parity condition on the creases that loop crosses.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

the flat pattern6 columns · 5 rowsthe course, closedradius 0.16676 sides of 0.1667 close into a ring of radius 0.1667, in the sheet's own unitsthe circle through the corners is 4.72% longer than the ring, which is what a 6-sided polygonowes its circle

The cylinder the pattern chooses

A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.

tessellation · Found patterns 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 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

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.

tessellation · Miura the pieces that are one panelleft and right edges identified — 9 pieces, 6 panels9 pieces on the drawing6 panels on the sheet10 creases, 4 verticeskeeps the sidetwo pieces of one shade are one piece of paper, a cell apart

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.

complexity · Map folding a square twist, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices16161616free letters40363632panels25202016V − 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

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.

tessellation · Twists 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

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.

design · Sheet shape Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubepackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable

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.

rigid · Deployables

Named alongside it

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

GluingBoundaryParityPeriodicityUnit cellCrease assignmentDesignManufacturingMiuraPanelPatchThe big-little-big lemma

All concepts