Origami & Folding
what one sheet can do
Concept

Surface area — where it appears

The area of a sheet, as against the volume or footprint it occupies. An organ whose job is to have area is solving a packing problem in reverse, and folding is what buys area inside a fixed envelope until the sheet's own thickness takes it back.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

How much surface fits in a body

An organ whose whole job is to have area — a gut, a gill, a cortex — is solving a packing problem in reverse. Folding buys surface inside a fixed volume, and with a sheet of zero thickness it buys an unlimited amount. With a thickness the curve turns over and then falls to nothing.

biology · Surface in a volume 02468101214010203040506070clearance above the basesurface, as a multiple of the basewalls stop at 40.0sheet 0.01, channel 0.05plies keep risingthey cross at 9.40the comb saturates at twice the reciprocal of its channel's share, and the stack does not saturate at all

The channel grows with what it feeds

A comb of standing walls beats a stack of plies by eight because its members do not share the clearance that pays them. Supply takes that back, and asymmetrically: a wall's channel has to be sized for the surface the wall carries, so it grows with the wall's height and is charged against the pitch, while a ply's channel is a constant charged against the clearance. The comb then saturates at twice the reciprocal of the channel's share, the stack does not saturate at all, and the two cross at a clearance the model gives in closed form.

biology · Surface in a volume the reach, the member length and the member count, against the anglesurface counted as a multiple of the base, lengths as multiples of the clearance90°60°30°10°1°the angle the members stand atthe reach: 200flat, at every anglehow long each member ishow many of them there areclearance 1, sheet 0.01 · the reach is 200 at every angle; the two factors move by 57

The angle the eight does not know

A comb's members are always drawn standing square to the base, and nothing has asked why. Lean one to an angle and it must be longer to reach the same clearance, which is more surface; it also takes more of the base to stand on, which is fewer members. The two are reciprocal and cancel exactly — the surface a comb holds is the same number from a right angle down to one degree, where each member is fifty-seven times the clearance long and there are two of them where there were a hundred.

biology · Surface in a volume two linings of one tube, and the ceiling neither passessurface per unit length of tube, from a sheet 0.01 thickradiuslayersfins at bestthe ceilinglayers over fins0.516078.51572.040016353146282.020022526125725132.01004100785027100532.0050the ceiling is the tube's cross-section over the sheet thickness, twice over, and no arrangement of flat sheet passes it

In a tube the standing members lose

Members standing across a clearance beat layers lying along it by eight, and every drawing of that argument has a flat base under it. Curve the base into a tube and the ranking inverts: radial fins converge, so the room they need is the room at their tips, and their best arrangement fills exactly half the cross-section. Concentric layers fill all of it. The eight becomes a half, and the half is exact.

biology · Surface in a volume 123456780.40.50.60.70.80.91how many lengths of finshare of the ceilingtips equally spacedheights halvingtwo thirdsradius 1, sheet thickness 0.01 · the share of the ceiling 2πR²/τ that fins of m lengths hold

The wedge belongs to one length

Radial fins inside a tube reach at best half of what any lining of sheet could hold, because converging fins leave empty wedges behind their tips. Tapering the fins cannot help: the tip already sets the count, and a fin cannot be thinner there than the sheet it is made of. Fins of several lengths can. Counted along the radius they are a staircase under a straight line, and the staircase with m steps is best with its steps equally spaced, where it holds exactly m ⁄ (m + 1) of the ceiling. The factor of two belonged to fins of one length, not to fins.

biology · Surface in a volume

Named alongside it

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

ScalingConstraintSurface in a volumeTrade-offIdealisationThicknessCorrugationPacking ratio

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