Optimality — where it appears
Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.
When symmetry costs
Design software and designers both reach for symmetry, and for a good reason: it makes the search enormously easier. It is a heuristic and not a theorem, and how much it gives away can be measured — including the case where the optimum is symmetric about an axis nobody imposed.
The square is a choice
Every packing on this site has been into a square, because origami paper is sold square. Hold the area fixed and vary the shape instead and the efficiency turns out to be spiky rather than smooth — with the same peak value at every proportion that is a ratio of two factors of the flap count, and nowhere else.
The largest triangle in a square
The biggest equilateral triangle a square sheet holds is tilted by exactly fifteen degrees and uses 46.4% of the paper. Both numbers come out of a quadratic — which means a compass reaches this optimum too, and folding's advantage is not needed here at all.
The biggest one that can also be folded
Which regular polygon uses a square sheet best, and which of them a fold can actually construct, are two questions with completely different pedigrees. Answered side by side over sixteen polygons, they turn out to agree — and the reason is that both are questions about the arithmetic of the same number.
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.
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.
The triangle a strip becomes
A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.
The proportion a band asks for
√2 is a shape: a rectangle either has it or does not, and what it buys is that halving returns the same shape. √3 is what a Möbius band needs, and it is a different kind of number — a minimum rather than a shape, with every longer strip working and no shorter one.
The square is in the answer
The largest regular polygon a square sheet holds is not increasing in the number of sides, and the octagon's win is the striking part: it uses 82.8% of the paper against the twelve-gon's 80.4% and the hexagon's 69.6%. Run the same census over rectangles and the octagon's advantage is gone — on every proportion tried the hexagon leads, and the order among the even-sided polygons reverses outright.
Every even polygon beats every odd one
Crossed with what each tool can build, the census of the largest regular polygon a sheet holds gives the same verdict on every proportion from a square to three to one: the best polygon is one a compass already builds, and the best polygon only a fold can build places fourth at best. The ranking itself stops moving at a proportion of 1.1284, where the hexagon overtakes the octagon. On every longer sheet only the short side holds a polygon, each polygon's share is a fixed constant divided by the length, and the constant — its area over the square of its least width — comes down to the circle's π⁄4 for even polygons and climbs up to it for odd ones. So every even polygon beats every odd one.
The sheet a polygon fits exactly
A regular polygon with 4k + 2 sides has flat edges along one axis and corners along the other, so there is one sheet, 1⁄cos(π⁄n) long, that it touches on all four edges at once. On that sheet it is beaten only by the multiples of four with fewer sides, and so it ranks exactly (n − 2)⁄4. That puts the fourteen-gon, which only a fold builds, third rather than fourth; the twenty-two-gon, which needs two folds at once, fifth rather than ninth; and the forty-six-gon, beyond two folds, eleventh. Seven sheets from a square to three to one had missed all three.
A price holds until the arrangement moves
Every edge of a subject's tree has a price — the scale lost per unit of extra length — and the obvious use of a price list is to spend a fixed total of limb where it is cheapest. Done a tenth of a unit at a time, re-pricing at every step, it works and then stops: the bird's scale rises 6.3 per cent in four steps and no further. But the prices do not hold while it happens. The bird's free tail stops being free after the first tenth, and its legs nearly treble in price without being touched. The lizard's prices hold for four steps, because its arrangement keeps the same three pairs at their limit for four steps. A price is a statement about which pairs are at or near their limit, and it lasts as long as they stay there.
Rounding in the cheap direction
A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.
Turning is uphill all the way
A regular polygon of 4k + 2 sides on a sheet a little longer than a square cannot lie flat: it turns, pressed against all four edges, until the sheet is exactly its own. Its share on the way has a closed form, and the closed form's slope is proportional to h² − 1 for every such polygon — flat on the square, rising all the way to the own sheet, and falling after it. So the own sheet is exactly the peak, the gain from the square to it is the average of one and the sheet's length, and the rank the census measured for polygons of this kind, (n − 2)⁄4, is now a theorem.
An odd polygon fits like its double
A regular polygon with an odd number of sides has a corner opposite every flat, so it looked as if the closed form for the turning polygons of 4k + 2 sides would need redoing with a different offset. It does not. An odd polygon is exactly as wide, in every direction, as the regular polygon with twice its sides — that polygon is the odd one averaged with its half-turn — and a box sees only widths. So the triangle fits wherever a hexagon fits, the heptagon wherever a fourteen-gon does, and on every sheet the odd polygon's share is its double's times 2 − sec²(π/2n). It peaks on its double's own sheet with the same gain; on a square it is tilted π/4n, the triangle's fifteen degrees; and on its own sheet it beats nothing but smaller odd polygons, never once passing the circle.
A sheet rewards the multiples of its sides
On a rectangle every odd polygon fits exactly as its double does, because a box measures widths and a width cannot see a half-turn. A triangle of paper has no half-turn, and the ranking it gives is a different kind of thing. On a regular sheet of m sides, a regular polygon beats the largest circle exactly when its sides are a multiple of m — it can then face every side of the sheet with one of its own and share the sheet's circle, taking n·tan(π/n) over m·tan(π/m) of it — and every other polygon to forty sides falls short. On a triangle that puts the triangle first, the hexagon second at two thirds, and the nonagon third: the first sheet on which a polygon only a fold can build takes a place on the podium.
The floor at degree four
A flat-foldable vertex creases some of the paper round it twice, and the ground it doubles had been found by search never to fall below the evenly spread vertex's — one band width squared at degree four. At degree four it can be proved. Kawasaki's condition leaves two free sectors; two identities of the kite turn the closed form into a function of their mean and half their difference; and two derivatives, each one line, show that parting the sectors always costs ground and that, with them equal, opening the mean to a right angle always saves it. So every flat-foldable degree-four vertex creases at least one band width squared twice, and only four right angles crease exactly that. The same pairing that makes the proof work is missing at degree six.
The rule has a width
On an equilateral triangle of paper the regular polygons that beat the largest circle are exactly the multiples of three. Tilt the apex a degree and every one of them loses share in proportion, while the circle loses almost nothing — a multiple's advantage is a corner and the circle's is a hilltop. So the rule holds only in a window, 44° to 79° of apex for the hexagon and 56° to 65° for the 24-gon, and past it the second place goes to polygons like the hendecagon that no single fold can build.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ConstructibilityInscribed polygonSheet shapeCircle packingSymmetryTotientRotational symmetryPacking efficiencyPaper proportionTree methodClosureConstraint