Origami & Folding
what one sheet can do
Concept

Efficiency — where it appears

How much of the sheet a design turns into the model rather than wasting. It is the quantity the tree method optimises, and it is one of several things a designer trades against each other.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 71% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs

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.

design · Circle packing 123456760%65%70%75%80%sheet, longer side to shorterfraction of the sheet claimedbest at 7.0 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

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.

design · Sheet shape 1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it

The corner is worth four times the middle

A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.

design · Sheet shape the pale bar is the solid sheet, the dark one the sheet with a holelower is better: it is the average share of a flap's disc that has to be paid forflap 0.060.9293 against 0.9496flap 0.10.8826 against 0.9137flap 0.150.8296 against 0.8714flap 0.220.7586 against 0.8159flap 0.30.6760 against 0.7496

A hole is cheap paper

A flap claims every point of paper within its own length of its tip, but only the paper that is actually there — so an edge is half price and a corner a quarter. A hole in the middle of the sheet is more edge, and holding the area fixed, a square with a hole in it is cheaper paper than a solid square at every flap length measured.

design · Sheet shape one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

A sheet with two edges

Design in this subject starts from a square, and the square's four edges are where every flap ends and every construction begins. A cylinder has two edges instead of four, and what a designer loses turns out to be measurable in the same letters a search counts — and what they gain is that the sheet is what gets manufactured.

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 1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes

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.

construction · Paper proportion 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 the most surface a lining can hold for each lumen, and the lumen that surface needsthe straight line is the demand; each dot is a lining that holds what its own lumen can supply00.2500.5000.750100.2000.4000.6000.8001share of the cross-section given to the lumenshare of the ceiling the fins hold1 length2 lengths3 lengths10 lengthsunlimitedleft of its dot a lining would hold more than its lumen can supply; right of it, the lumen is wider than its surface needs

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.

biology · Surface in a volume the surface a supplied lining holds when its supply has to reacheach unit of surface at radius r needs a channel 1 + λ(r − ρ) times as wide as at the lumen's edge00.2500.5000.7501-2-101log₁₀ of the channel's width per unit of surface, in sheet thicknessesshare of the ceiling the fins holdone lengthunlimited lengthsthe same, reach freethe upright line is the channel of a fifth of a sheet per unit of surface; its dot is the unlimited lining there

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.

biology · Surface in a volume

Named alongside it

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

Sheet shapeCircle packingOptimisationBoundaryDesignGluingSurface areaSurface in a volumeTrade-offDesign techniqueFlapThe Möbius band

All concepts