Origami & Folding
what one sheet can do
Series

Surface in a volume — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · biology
  2. 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

    Nothing grown has a seam

    A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.

    part 2 · biology
  3. 0204060801000510152025foldssurface heldno supply — 50 foldsδ = 0.01 — 25 foldsδ = 0.03 — 12 foldsδ = 0.09 — 5 foldsbox of side 1 · sheet thickness 0.01 · optimum at S ⁄ 2(t + δ), so supply and sheet are charged the same way

    The surface has to be supplied

    The curve that turns over does so because the sheet's own thickness fills the box it is folding into. A surface in a body has to be reached as well as fitted, and the channel that reaches it takes depth out of the same box on exactly the same terms — so the best fold count and the surface it delivers both fall by the ratio of the sheet's thickness to the sheet and its supply together.

    part 3 · biology
  4. 0204060801000510152025foldssurface held1 surface — 50 folds2 surfaces — 25 folds3 surfaces — 17 folds4 surfaces — 12 foldsbox of side 1, thickness 0.01 · a ceiling goes as depth², so m sharers of one depth reach one m-th of it between them

    Two surfaces in one box

    A body folds several surfaces into one volume and each of them does a different job. Dividing the depth between them looks like a fair split costing nothing overall, and it is not: the area a single surface can reach goes as the square of the depth it has, so m surfaces sharing a depth reach a total of exactly one m-th of what one of them would have reached alone.

    part 4 · biology
  5. -2.5-2-1.5-1-0.5050100150200250depth given to the inner level, log₁₀ of the boxsurface over the flat sheetone level, the whole boxreaches 250.0two levels, any splitnever above 62.5box depth 1 · sheet 0.001 · one level reaches 250.0, and a nest of two reaches 62.5 however the depth is shared

    A nest pays four a level

    A corrugation folded inside the panels of another looks like the arrangement that multiplies surface rather than dividing it. It multiplies the factors and divides the depths, and the depths cancel: a packed level can hold at most its depth over four times what it folds, the thing it folds is as thick as the depth the level below was given, and so a nest of L levels reaches at most the box over 4ᴸ sheet thicknesses. One level with the whole box beats any nest of two by exactly four.

    part 5 · biology
  6. 012340200400600800clearance above the basesurface, as a multiple of the basewalls: 2c ⁄ τplies: c ⁄ 4τeight times lesssheet thickness 0.01 · both lines are straight and their ratio is eight everywhere, so no clearance makes the stack competitive

    Standing up beats lying down by eight

    A level that fills its clearance with plies lying flat makes every ply share the clearance, and the sharing costs a factor of four. A level that fills the same clearance with walls standing on the base gives every member the whole height and charges them only for footing. The ratio is exactly eight, at every clearance and every thickness — and it is the difference between a cost charged against depth and a cost charged against the space beside it.

    part 6 · biology
  7. 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.

    part 7 · biology
  8. 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.

    part 8 · biology
  9. 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.

    part 9 · biology
  10. 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.

    part 10 · biology

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