Origami & Folding
what one sheet can do
Series

Grafting — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the design as it stood20.00 of paper · 5 creases meet the cutthe same design, one strip wider21.80 of paper · the strip is 0.45 acrossthe band is 0.45 × 4 = 1.8000, and that is the entire difference between the two patternsall 44 creases away from the cut keep their length; the 5 that cross it are longer by 0.45 and by nothing elsea design grows by accretion because the arithmetic of growing it is this short

    Paying in paper

    A feature added to a finished design costs exactly the paper inserted for it. Cut the crease pattern along a line, slide in a strip, and every existing crease continues across it unchanged — so the bill is the strip's width times the length of the cut, and there is no second term.

    part 1 · design
  2. 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.

    part 2 · design
  3. the lines a graft could useevery gap between vertex columns, and every gap between vertex rows, triedpatternvertical lineshorizontal linesgraftsThe preliminary base0 of 20 of 2nowhereThe Miura fold6 of 130 of 4one wayThe square twist0 of 60 of 6nowhereThe hexagon twist0 of 42 of 11one wayThe Yoshimura pattern0 of 120 of 5nowhereFold and cut — the triangle2 of 82 of 8both, in margin onlyThe tapered corrugation7 of 150 of 4one wayThe waterbomb tessellation0 of 80 of 8nowherea 4 by 4 grid4 of 44 of 4both waysthe grid, one square creased diagonally3 of 43 of 4both waysa line is admissible when every crease it crosses is square to it; the entry is admissible of the lines between vertex columns or rows

    A graft needs a square line

    A strip can be slid into a finished crease pattern only along a line every crossed crease meets square, because only such a crease continues across the strip as itself. Tried on every line between the vertex columns and rows of eight printed patterns, four take no strip in either direction and three take one in a single direction. The eighth, the fold-and-cut triangle, appears to take strips both ways, and every line it admits runs through blank margin. No printed pattern takes a strip across its creases in both directions — and one diagonal crease in a plain grid removes exactly the row and the column it sits in.

    part 3 · design
  4. 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.

    part 4 · design
  5. a 6 by 6 grid, and the lines it still admits a strip onthe same diagonals placed three ways — the count is of lines, not of creases05100123456diagonal creases in the patternlines a strip could be slid into, both directionsno two in a row or a columnall in one rowdropped at randoma diagonal costs the row and the column it sits in, so a design that keeps its diagonals in a few rows keeps its lines clear

    A design that keeps its lines clear

    A strip can be slid in only along a line every crossed crease meets square, so a diagonal crease spends the lines it crosses — and the census of eight printed patterns found four taking no strip in either direction. What decides how fast a design spends them is not how many diagonals it has but which rows and columns they sit in: six diagonals on a six-by-six grid leave twelve clear lines when they share a row and none at all when no two do, from the same six creases and the same amount of paper.

    part 5 · design
  6. 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.

    part 6 · design
  7. five features, rounded up to whole spacings — the extra paper, as a share of the bill as askedstrips 0.08, 0.14, 0.2 across and 0.3, 0.06 down, on a unit sheet; each row is one grid4 by 4121.5%0.25 + 0.25 + 0.25 across, 0.5 + 0.25 down5 by 567.5%0.2 + 0.2 + 0.2 across, 0.4 + 0.2 down6 by 661.1%0.167 + 0.167 + 0.333 across, 0.333 + 0.167 down7 by 757.8%0.143 + 0.143 + 0.286 across, 0.429 + 0.143 down8 by 854.4%0.125 + 0.25 + 0.25 across, 0.375 + 0.125 down9 by 933.9%0.111 + 0.222 + 0.222 across, 0.333 + 0.111 down10 by 1018.1%0.1 + 0.2 + 0.2 across, 0.3 + 0.1 down11 by 1134.0%0.091 + 0.182 + 0.273 across, 0.364 + 0.091 down12 by 1220.8%0.083 + 0.167 + 0.25 across, 0.333 + 0.083 down13 by 1321.4%0.154 + 0.154 + 0.231 across, 0.308 + 0.077 down14 by 1422.7%0.143 + 0.143 + 0.214 across, 0.357 + 0.071 down15 by 1523.1%0.133 + 0.2 + 0.2 across, 0.333 + 0.067 down16 by 1623.3%0.125 + 0.188 + 0.25 across, 0.313 + 0.063 down17 by 1734.1%0.118 + 0.176 + 0.235 across, 0.353 + 0.118 down18 by 1825.3%0.111 + 0.167 + 0.222 across, 0.333 + 0.111 down19 by 1917.5%0.105 + 0.158 + 0.211 across, 0.316 + 0.105 down20 by 2010.6%0.1 + 0.15 + 0.2 across, 0.3 + 0.1 down10 by 10 costs 18.1 per cent, and every one of the next 8 finer grids costs more than it does

    A spacing costs one plus the other side

    Rounding a grafted strip up to a whole number of grid spacings is the standard remedy for a width the grid cannot hold, and with strips in one direction it costs the rounding times the cut. With strips both ways it costs more, because the crossing term is the product of the two families' totals: a spacing added across is charged once against the sheet and again against every strip running down. The same arithmetic makes the bill uneven in the grid — ten divisions beat every grid from eleven to eighteen on one design — and puts two families exact on their own grids at their least common multiple.

    part 7 · design
  8. 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.

    part 8 · design
  9. 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.

    part 9 · design

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