Origami & Folding
what one sheet can do
Concept

Design cost — where it appears

What a feature takes from the sheet, measured rather than estimated: the paper inserted for it, the grid it divides, the flaps it displaces. A cost of this kind is stated for one operation and composes when several are applied, which is why it is worth computing exactly for the single case.

Named by 3 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 a 8 by 8 grid, spacing 0.125 — what each strip width does to itthe bar is how many times finer the grid becomes — one means unchanged0.1251x1 spacings — the grid is unchanged0.251x2 spacings — the grid is unchanged0.3751x3 spacings — the grid is unchanged0.06252xthe grid becomes 0.0625, 2 times finer0.15xthe grid becomes 0.0250, 5 times finer0.1325xthe grid becomes 0.00500, 25 times finer0.25xthe grid becomes 0.0250, 5 times finera width four per cent away from a spacing divides the grid by twenty-five; a width half a spacing away divides it by two

The width is charged in grid

A grafted strip may be any width at all and the bill in paper is exactly width times length, with no second term — which is what the first of these essays established, and is a statement about area. The grid is a different property of the same pattern and it is not conserved: a strip whose width is not a whole number of spacings puts every vertex past the cut onto a finer grid, and a width four per cent away from a spacing divides the grid by twenty-five where a width half a spacing away divides it by two. A width nearly right costs far more than one plainly wrong.

design · Grafting

Named alongside it

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

Box pleatingCrease patternGraftingAreaConservationDesign constraintGridRiver

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