Origami & Folding
what one sheet can do
Concept

Area — where it appears

How much paper a pattern or a feature occupies, counted over the panels the creases enclose rather than assumed from an outline. It is the quantity folding conserves exactly — a folded sheet has the same area it started with, redistributed — which is what makes it the currency a design operation can be priced in.

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

one graft, then a second across itstrips 0.3 and 0.3 wideas designedone stripand one across itthe first strip adds 0.300the second adds 0.390two separate bills would be 0.600the sheet gained 0.690the excess is 0.090and the strips cross over 0.090the crossing rectangle is 13.0 per cent of everything the two features cost

The second term

A feature grafted into a finished design costs exactly the paper slid in for it, and the bill has one term. Add a second feature across the first and it has two: the rectangle where the strips cross is paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.

design · Grafting several strips each way, and what the sheet gainedwidths 0.14, 0.14, 0.14 across and 0.18, 0.18 down3 strips across, totalling 0.422 strips down, totalling 0.36charged separately: 0.780the sheet gained: 0.931the excess: 0.1512the two totals multiplied: 0.15126 rectangles where they crossthe crossing term is 16.2 per cent of everything the features cost, and it is one term

Six rectangles and one term

Two grafted strips crossing leave one rectangle both features are charged for and neither uses. Three strips crossing two leave six, and the obvious budget adds them up. It does not have to: the six rectangles sum to the product of the two families' total widths, exactly, so a design with any number of features is priced by two numbers rather than by a double sum — and a family of strips in one direction alone carries no crossing term at all, however many of them there are.

design · Grafting the paper one strip costs over its width as asked, on each grid, with and without latitudedashed: rounded up; solid: rounded down within the latitude first; the vertical marks where the solid curve crosses zero-0.05000.0500.1004102030405060divisions across the sheetextra paper a strip, fraction of the sheetrounded upwith latitudelatitude 0.020 of the sheet: the break-even grid is 25.0 divisions

A grid chosen before the features

A designer fixes a box-pleating grid before knowing what will be grafted into it, so the question is what a grid costs in expectation. For widths nobody chose — spread evenly over many spacings — rounding up costs exactly half a spacing a strip on every grid, prime or composite: the sawtooth of any one design averages away, and the familiar bound on the rounding bill is twice its expected value. Divisors earn a discount only when the widths are themselves simple fractions of the sheet, and then 24 and 48 beat 16 and 32. A latitude changes the question entirely. Allowed to fall short by δ, every strip saves exactly δ on average, so the grid 1/(2δ) holds features at no expected cost — but a particular design keeps its sawtooth.

design · Grafting one budget, one split, two sheets: the bill for the five featuresthe arch is any rectangle, the valley a square sheet with a margin on the shorter side0.6000.800100.2500.5000.7501share of the budget spent on the strips acrosspaper over the original sheetany rectanglesquare sheeta shared budget of 0.100 of the sheet

The split that is worst on a rectangle

Two families of grafted strips share an allowance of shortfall, and the bill is a product of two sides whose sum the allowance fixes. So where the allowance goes is a question about a rectangle with a set perimeter: its area is largest when level and smallest when lopsided. On a sheet that may be any rectangle the split that levels the sides is the dearest there is, and it is barely dearer, because the gap enters squared. On a square sheet the same split is the cheapest, and the gap enters to first order, ninety-four times as heavily on five features. On a grid the first answer dissolves — the strip nearest a line decides, not the family — and the second does not.

design · Grafting

Named alongside it

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

Box pleatingDesign costGraftingCrease patternDesign constraintGridConservationRiver

All concepts