Origami & Folding
what one sheet can do
Concept

Packing efficiency — where it appears

The share of a sheet that ends up as useful flap rather than as waste. It is an open problem for most flap counts, so the numbers quoted are the best any search has found rather than optima.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

2 discs53.9%r = 0.29293 discs61.0%r = 0.25434 discs78.5%r = 0.25005 discs67.3%r = 0.20716 discs66.3%r = 0.18767 discs66.9%r = 0.17448 discs72.8%r = 0.17029 discs78.5%r = 0.1667hexagonal density 90.69%every bar is the best a seeded search found, not a proved optimum —which is the honest state of the problem for all but the first few values of n

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.

design · Circle packing 7 flaps, packed as tight as they will goradius0.174457630loose flaps1room to move0.1127the inner ring is where that flap's centre may sit; the algorithm reports one point of it and stops

The flap nobody holds

An optimal packing is presented as an answer: here are the circles, here is where they go. For some numbers of flaps that is not what it is. The best arrangement of seven discs in a square leaves one of them free to wander over an eleventh of the sheet without changing the answer at all — and the algorithm reports one point of that region and stops.

design · Circle packing 23456789101105101520flapshow much the requirement costs, per cent14.61.70.0-0.05.60.510.810.44.72.8

The shapes the optimum has

Requiring a circle packing to be its own mirror image halves the number of coordinates a search has to find, so the same effort covers a much smaller space. Whether that helps depends on something the search cannot know in advance — whether the best packing was symmetric — and measured flap count by flap count the answer alternates without a pattern anybody could use.

design · Symmetry two tips 0.6 of a sheet width apartthe bar is how long a flap each can carrysolid paper0.300the discs meet halfwaya slot 0.1 wide0.35719% further than solid papera slot 0.2 wide0.40435% further than solid papera slot 0.3 wide0.46655% further than solid papera slot 0.4 wide0.50067% further than solid papera slot 0.5 wide0.55184% further than solid paperthe tips do not move; only the paper between them does

Spending the cheap paper

A hole makes a sheet cheaper by the square inch, because a flap against its rim claims only half a disc. Whether a design can spend that was left open, because no packing search here could express a region that is not paper. It can now, and the mechanism turns out not to be the discount at all: across a hole, two flaps may overlap, and two tips three fifths of a sheet apart carry flaps of 0.500 rather than 0.300.

design · Sheet shape the bar is how much cheaper the paper is than a plain square of the same areaflaps of 0.061.40%notch 0.60% · hole 1.40%flaps of 0.12.73%notch 1.31% · hole 2.73%flaps of 0.154.59%notch 2.28% · hole 4.59%flaps of 0.226.76%notch 3.20% · hole 6.76%flaps of 0.39.85%notch 4.54% · hole 9.85%same paper removed, twice the saving — a hole has four sides of rim and a notch has three

A notch is not a hole

Remove the same rectangle of paper from the middle of a square and from its edge, and the two sheets are not worth the same. The hole is cheaper paper at every flap length measured — 4.71% cheaper than a plain square of equal area against the notch's 2.58% — because what a cut is worth is rim with paper on both sides of it, and a notch spends one of its four sides on an edge the sheet already had.

design · Sheet shape a flap of 0.12 of the sideless is cheaper: a flap claims only the paper that is actually therethe open middle of the sheet1.0000 of the discagainst an edge0.5000 of the discin a corner0.2500 of the discagainst the hole0.5000 of the discin the hole's outside corner0.7500 of the disc

The corner premium, with no corners

A square's corners are its most valuable paper: a flap placed there is claimed by a quarter-disc rather than a whole one, so the corner goes four times as far as the middle. A closed sheet has no corners at all, and the accounting that ranks sheet shapes by their corners has nothing left to rank.

design · Sheet shape fins of several lengths with a lumen kept clear down the middlethe diagonal is the most fins the circumference allows at each radius; the staircase is what the fins achieveradius, axis to wallfree lumenlumen radius 0.40R: 16% of the cross-sectionfins of 3 lengths hold 72.0% of the ceilinga lumen out to 0.25R would have cost nothingthe most any lining could hold now: 84.0%

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.

biology · Surface in a volume how far each search is from the best packing knownboth at 120 restarts of 6,000 steps; nothing below zero is possible02461011121314151617181920discsshort of the best packing known, per centthe free searchbest: a mirrorbest: a half-turnbest: a quarter-turn

Equal effort was counted in steps

A circle-packing search required to be symmetric was found to beat a free search of the same effort, and the reason given was that a smaller space is easier to search. Past eleven discs the test gets sharp: the free search falls behind the best packings known, 0.13 per cent at ten discs and 6.07 at twenty, and a symmetric search leads it by as much as 4.18. But tighten the free search's packings to the arrangement they were heading for and most of that shortfall disappears — at twenty discs, all of it. Give the free search twice the steps and the lead reverses. The effort was equal in steps and unequal in what a step does: a symmetric search moves half as many coordinates, so each is moved twice as often.

design · Symmetry

Named alongside it

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

Circle packingOptimalitySheet shapeTrade-offConstraintDesignDesign spaceEfficiencyFlapOptimisationSymmetryTree method

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