Origami & Folding
what one sheet can do
Concept

Flap — where it appears

A point of a folded model that sticks out — a leg, a wing, a horn. Its length costs a disc of paper whose radius is that length, which is the conservation law the whole tree method is built on.

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

Levery point within L is spentthe flapLL = 0.28 of the sheet's side, so the disc costs πL² = 24.6% of itthe circle is not a metaphor — it is the paper the flap consumesso designing a base is packing circles

A flap costs a circle

A flap of a given length uses up every point of the sheet within that distance of it. Two flaps whose circles overlap are asking for the same paper twice — and that one observation turned origami design from an art into an algorithm.

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 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 a flap of 0.12 of the sheet's side, on two sheets of the same areadarker is cheaper: less of the flap's disc is paper that has to be paid forwith a holesolid, same areamean claim 0.8646mean claim 0.8959

A base needs an edge to point at

Every flap of a uniaxial base ends in a point, and every point is made of paper that came from the sheet's boundary. A sheet with no boundary has nowhere for a point to come from, and a sheet with half a boundary can only have points at the half it has left.

design · Uniaxial bases a body with four legs and a tail — one internal edge of 0.8circles of radius m·ℓ, and every pair's requirement drawn — the ones through the body ask for more than the two circles do

Every pair, not every circle

A uniaxial base is designed by packing a circle for each flap, and the circles are not the condition. The condition is that every pair of the subject's extremities be separated on the sheet by the distance between them through the tree — which for two flaps meeting at one point is the sum of their lengths, and for two flaps across a body is more. Circles are the case with no body in it, so a design read off circles alone is promised a base sixteen to thirty per cent larger than the sheet can give.

design · Uniaxial bases a body of three segments — two internal edges, 0.6 and 0.9the segments are the pairs at their limit — 4 of 8 of them measured through the body rather than around it

What the condition does not decide

A tree of seven leaves imposes twenty-one separations and eight of them bind. The rest are slack, the eight pin six of the seven leaves against the sheet's own edges, and the seventh can be moved half a per cent of the sheet for nothing. The requirement that looks quadratic is doing linear work, and what it leaves undecided is the part a designer is actually choosing.

design · Uniaxial bases what lengthening each edge of a body with wings, legs, a head and a tail coststhe scale falls from 0.2651 by this much per unit of extra length, measured by re-solving the arrangementchest–head0.0786a flap, 5.9% of the scale per 0.2rump–ll0.0471a flap, 3.6% of the scale per 0.2chest–rump0.0429the body, 3.2% of the scale per 0.2chest–wl0.0361a flap, 2.7% of the scale per 0.2chest–wr0.0264a flap, 2.0% of the scale per 0.2rump–tail0.0123a flap, 0.9% of the scale per 0.2rump–lr0.0120a flap, 0.9% of the scale per 0.2an edge no tight pair passes through is an edge the design can spend freely, and the condition says which

The price of a limb is not its length

Lengthening an edge of a subject's tree costs the design some of its scale, and the amount can be measured by re-solving the arrangement. It is not proportional to the edge, and it is not the body that is dearest. On a bird whose wings are twice its legs, the head — nine tenths of a unit against the wings' one and six — costs three times what a wing costs, and two edges of a lizard cost nothing at all.

design · Uniaxial bases

Named alongside it

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

DesignCircle packingTree methodUniaxial baseBoundaryConstraintGluingSheet shapeEfficiencyPacking efficiencyConservationContact

All concepts