Grafting — where it appears
Named by 9 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Box pleatingCrease patternDesign costGridAreaDesign constraintDesign techniqueRiverConservationCircle packingDesign spaceUniaxial base