Optimisation — where it appears
Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.
Packing is the hard part
Once a subject is a set of circles, designing the model is fitting them into a square. That step has no general algorithm, no known optimum, and it is where every remaining difficulty in origami design now sits.
How much paper is wasted
The efficiency of a design is the fraction of the sheet its flaps can claim, and for almost every number of flaps nobody knows the best possible value. The bars in these figures are the best a search could find, which is not the same thing.
Four questions about one sheet
Deciding, counting, listing and optimising are not four difficulties of one problem. They are four problems, and folding is the subject that proves it: a ruled map is trivial to decide and unsolved to count, while a general crease pattern is the other way round.
Getting close instead of getting it right
When the best answer is out of reach the question stops being what it is and becomes how much is lost. For packing discs into a square the loss is measurable: a seeded search in this repository comes within a fifth of a percent of the best radius anybody has proved, and proves nothing.
The molecule that does not exist
The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.
What the grid settles
Box pleating is usually defended as a trade: give up efficiency, buy creases that land where they should. There is a second thing it buys and nobody quotes it — on a lattice the best possible packing is a finite question with an answer, while off the lattice nobody knows the best packing of six circles in a square and probably never will.
Spelling a tree on a grid
Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.
The skeleton changes its mind
The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.
Two packings, one radius
A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.
The grid belongs to the subject
Rounding a subject's limbs to whole grid units trades shape for size, and on any one grid the cheap-direction rounding is the largest. Put every rounding on every grid from three to sixteen units side by side and only a handful are worth having. For the bird, nine of 121 roundings are beaten by nothing on both counts; they come from seven grids, starting with the exact spelling on sixteen and buying size at about a quarter of a per cent for each point of worst-limb error. Eight, twelve, eleven, four and three units contribute nothing — every rounding on them is beaten by one on another grid. For the lizard, eight and twelve are on the frontier and six, ten and fourteen are not. No grid is good in general; a grid is good for a subject.
A lumen costs nothing until it does
A tube lined with radial fins of m lengths holds at best m ⁄ (m + 1) of the most surface its sheet could carry, with the fins' tips on equally spaced rings. A tube also has to carry something down its middle. Keep a central lumen of radius ρR clear and the best lining puts its first ring of tips at the lumen and spaces the rest evenly to the wall, holding 2ρ(1 − ρ) + (1 − ρ)²(m − 1) ⁄ m of the ceiling — until the lumen is smaller than R ⁄ (m + 1), where it costs nothing at all, because the fins' own optimum already leaves that much empty. Past it, each further unit of lumen costs surface at a rate rising from nothing towards one. With unlimited lengths the frontier is the straight line 1 − ρ²: the bend is the price of having few lengths, and the free lumen is the wedge they wasted.
Supply decides whether lengths pay
A tube lined with fins has a frontier of surface against the lumen left down its middle, and a lining that must supply its own surface needs a lumen that grows with the surface. The two meet at one point per number of fin lengths. When a channel is narrow for what it serves, the meeting is on the flat stretch and supply costs nothing; when it is wide, every lining meets near the full lumen, and the lengths that were worth half again are worth under one per cent.
Reach taxes the extra lengths
A tube's fin lining that must supply its own surface was priced with a channel of the same width for every unit of surface, wherever it stood. Charge instead for how far the supply has to carry, and the extra lengths of fin — which multiply toward the wall — pay more for each unit of surface than a single length does. Where supply was cheap they are worth half again rather than two thirds; the passage to lengths buying nothing moves to channels half as wide; and where supply was dear, nothing changes at all, because a thin ring has no reach inside it.
The worst angle sets the price
On a triangle of paper that is nearly equilateral, the largest regular hexagon loses 0.448 percentage points of the sheet for every degree the triangle is out of true, the nonagon 0.267 and the dodecagon 0.193. Those rates were measured. They have a closed form, 2n tan²(π/n) / 9√3 a radian, and it depends only on the worst of the triangle's three angles, whatever its shape. Rate times sides falls to 2π² / 9√3, and the window in which each polygon beats the circle shrinks as ninety degrees over its number of sides.
A shared supply is the same bill
A tube's fin lining charged for how far its supply has to carry was expected to do better under a supply that branches, one vessel running out along each fin and feeding everything it passes. It does not, and it cannot: with channels sized to the flow they carry, the per-unit charge and every branching tree are the same integral. Sharing pays only when a channel's width grows more slowly than its flow, as Murray's and Poiseuille's vessels do, and then branching buys back exactly what those economies would otherwise have charged, to within a fifth of a point.
A start is worth an inverse square
Straight tucks gathering a flat disc into a cap leave a worst strain that falls as the starts multiply, and every essay on them counted starts without saying what one buys. It buys an inverse square. A chord of the hiding curve misses it by its width squared times the curve's bend over eight, so starts placed for strain leave a worst strain of I² ⁄ 8m², with I one integral of the cap's shape — 1.464 for a hemisphere. A paper that gives ε therefore needs about I ⁄ √(8ε) starts, a tenth of the give costs three times the starts, and letting the hiding cross the curve saves exactly a factor of √2. The same argument says why the starts spread outward, and by how much.
The window has an exact edge
How far an isosceles triangle may stray from equilateral before its largest regular hexagon falls behind its inscribed circle was measured — 15.6 degrees closing the apex, 19.1 opening it — and estimated to first order as the polygon's lead over its rate. Neither number needs measuring. The fitting problem has three sides and three unknowns, so its bound is an equality, the polygon keeps a side on the base, and its share is one line. That line carries the same factor as the circle's, so the circle's own loss cancels out of the comparison entirely, and each edge of the window is an angle in closed form. The opening half is wider because opening the apex lengthens the perimeter. And the window does not shrink as ninety degrees over the sides, as the first-order estimate says, but as (2 − √2)·180° over them — about 105 — because the corner's reach bends back as fast as it rises.
Two sides at sixty are enough
On a nearly equilateral triangle every multiple of three beats the inscribed circle, and to first order how long it keeps doing so depends only on the worst of the triangle's angle errors, which makes the window a hexagon in the plane of two errors. Fitted exactly, the window is a scalloped star. It reaches furthest — 21.8 degrees for the hexagon, against 15.6 and 19.1 on isosceles triangles — where one angle opens and another closes with the third left at sixty, because two sides still meet at sixty and the polygon can lie flush against both. Far outside it, wherever all three sides lie along the polygon's own directions, the polygon wins again.
Named alongside it
The objects these essays reach for when they reach for this one.
Trade-offTree methodEfficiencySurface areaSurface in a volumeBox pleatingCircle packingGridToleranceThe decision problemDesign techniqueEnumeration