Generator

How much surface fits in a body — the surface in volume generator

A generator in the folding nobody designed library, called 81 times across 17 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

surface-in-volume is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

`view: "comb", show: "steps", ms: [1, 3]`

`view: "comb", show: "slivers"`

`view: "comb", show: "lumen", lengths: 3, lumen: 0.4`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

3 sharing arrangements: one surface reaches 250.000 and 4 sharing reach 62.500 between them ×5
servicing the surface moves the best fold count from 50 to 5 ×5
the best whole fold count sits within 0.50 of S ⁄ 2(t + δ) on all 4 channel depths — a channel is charged exactly as the sheet is ×5
the surface a corrugation fits into a box of side 1 is computed at 8 fold counts, at a thickness of 0.01 ×5
1 of the counts fill the box with sheet alone and hold no surface at all ×3
one level of whole folds reaches 250.00 against the closed form D ⁄ 4t = 250.00 ×3
the deepest channel drawn costs 90% of the surface the box could otherwise hold ×3
the surface peaks at 64 folds and falls after it — the thickness, not the geometry, is what stops it ×3
and every level added makes the best nest worse, from 250.0 to 3.84 ×2
at every one of 48 depths given to the inner level the pair reaches at most D ⁄ 16t = 62.5 — the inner depth cancels out of the ceiling ×2
each level stays under its own ceiling, and the nest reaches 15.6 against D ⁄ 64t = 15.6 ×2
each of the 3 arrangements drawn reaches the same 200, although the members differ in length by a factor of 2.9 ×2
filling the gaps between long fins with shorter ones recovers a real part of what the wedges waste — the best schedule drawn holds 443 against one length's 314 ×2
no nest of any depth tried rises above D ⁄ 4ᴸt — 250.0 at 1, 62.5 at 2, 15.6 at 3, 3.91 at 4 ×2
one level with the whole box reaches 250.0, which is 4.00 times the best nest of two ×2
the comb has no peak at all: it rises to 100 walls and then stops, because one more wall has nowhere to stand ×2
the total surface across m sharers is the single-surface ceiling divided by m, to 0.0% ×2
a comb of walls reaches exactly eight times what a stack of plies reaches in the same clearance out of the same sheet — two for the wall's second face, four for the clearance the plies have to share ×1
a comb standing inside a tube is best at a height of exactly half the radius, and what it reaches there is the tube's own cross-section divided by the thickness of the sheet — 314 at radius 1 ×1
a supplied comb cannot pass 40.0 however tall its walls are, and by the right of this figure it has reached 98.7 per cent of it ×1
a taller wall serves more surface and needs a wider channel for it, so the share of the bare ceiling a supplied comb reaches falls at every height — 44%, 29%, 17%, 9%, 5% ×1
across clearances from 0.4 to 4 and thicknesses from 0.002 to 0.05, the ratio is eight on every row ×1
and a fatter channel brings the crossover nearer — 0.02 at 11.5, 0.05 at 9.40, 0.1 at 8.70, 0.2 at 8.35 — so the better a surface is supplied the deeper a body may go on standing walls ×1
and at every height it holds exactly one minus the height over the radius of what the same area of flat base would hold, so the loss is the fraction of the tube the members have eaten ×1
and it is the product that is fixed rather than either factor: length times members is the same number at every angle, which is what the sine cancelling means when it is drawn instead of written ×1
and none of them reaches the ceiling of 628, because a fin of any length still leaves a wedge behind its own tip ×1
and the ceiling is met exactly at 2 of those depths, so it is the answer rather than a bound nobody reaches ×1
and the factor is exactly the cross-section the fins fail to occupy: at their best height they fill half of it and leave the wedges between them empty, at every radius ×1
and the stack at most 0.16 per cent — neither ceiling is an artefact of allowing fractional members ×1
at 4 numbers of lengths and four lumen radii each, no random schedule of tips beats the pinned staircase, and its share is 2ρ(1 − ρ) + (1 − ρ)²(m − 1) ⁄ m once the lumen passes R ⁄ (m + 1) ×1
at a reach cost of 8, a vessel per fin always holds less than a branched supply, and the five routings and sizings differ by at most 1.6 points where reach itself costs the lining 18.8, 18.7, 14.5, 3.2 ×1
at k = 0.2 and a reach cost of 2, one length's surface stands on average 0.250 of the radius from its lumen's edge and unlimited lengths' 0.285 — the extra lengths stand further out, where reach costs most ×1
at k = 1 one length holds 88.9%, 91.5%, 94.3% of what unlimited lengths hold as the reach cost rises through 0, 2, 8 — dearer reach makes lengths worth less ×1
every vessel of a branched supply carries at least its own fin's surface beyond it, 2(1 − r); unlimited lengths' vessels carry (1 − r²) ⁄ r, which at a lumen of 0.3 is 2.17 times that, and 2 lengths' vessels peak at 2.22 ×1
for the supplied lining of 3 lengths at k = 1 and a reach cost of 2, the charge each unit of surface pays for its own reach and the channel a branching tree needs to carry everything beyond each radius enclose the same area, 0.1170 and 0.1170, against 0.1170 from the lining's bill ×1
for this sector of two lengths, vessels sized by flow to the power 0.50 cost 84.1% as much forked from the long fins as fed one to a fin ×1
four thousand random schedules of two, three and four lengths each hold less than equal spacing does ×1
halfway from the lumen to the wall the flow-sized charge has risen half its full amount; with Murray's vessels 37% of it and with Poiseuille's 29% — economies of scale move the charge toward the wall, where the vessels are thinnest ×1
inside a tube the concentric layers hold twice what the radial fins hold at their best height, and the excess over two is the half-layer a whole count leaves over — it halves every time the tube doubles ×1
leaning the members over changes the surface a comb reaches by 3e-14 of it — from a right angle down to one degree, where each member is 57 times the clearance long and there are 1.75 of them per unit of base instead of 100 ×1
nesting a comb inside a comb never reaches what one comb reaches — 195 against 200 — and it approaches it only by spreading the outer walls until they multiply nothing ×1
of the three heights drawn the middle one holds the most, and it is the one at half the radius ×1
on the unpinned optimum the widest gap between neighbouring fins, anywhere outside the lumen, is exactly one sheet thickness, whatever the number of lengths — it is just behind the first ring of tips ×1
past a demand of one half, what fins of one length hold is exactly 1 − 1 ⁄ (1 + 2k)² of what unlimited lengths hold, so the most further lengths can add falls as the inverse square of the demand ×1
priced so the single length pays what it paid, no routing or sizing moves unlimited lengths by more than 1.71 points from the flow-sized lining, where reach itself costs them up to 19.6 points, at a channel of 0.31; and at every channel a vessel per fin holds less than a branched supply ×1
rounding the member count to a whole number costs the comb at most 0.00 per cent ×1
successive halving stays below two thirds of the ceiling however many lengths it uses; equal spacing passes 89 per cent at 8 ×1
the fins counted one length at a time and the staircase under the line give the same share at every number of lengths, and equal spacing gives m/(m + 1) exactly ×1
the pair's reach is the single comb's times one minus the share of the pitch its own walls occupy, at every pitch drawn ×1
the ratios of each level's depth to what it folds multiply to exactly D ⁄ t = 1000.0, however the depths are chosen ×1
the stack is best at 50.0 plies and gets worse on both sides of it — half of them would be as good as twice ×1
the two architectures are equal at 7τ ⁄ β + 8 for every channel width drawn, which is where the closed form says and is checked against both curves rather than read off them ×1
with 3 lengths and a lumen of radius 0.40R, fifteen hundred random schedules of tips outside the lumen all hold less than tips at ρ then equally spaced ×1
with 3 lengths at k = 1 the tips are equally spaced from the lumen to the wall while reach is free, and with reach costing 2 or 8 the gaps between them widen toward the wall — 0.082, 0.086, 0.092; 0.058, 0.065, 0.078 ×1
with 400 lengths the share is within three thousandths of 1 − ρ², the cross-section the lumen leaves, at every radius: in the limit a unit of lumen costs a unit of surface ×1
with a channel 0.2 sheet thicknesses wide for each unit of surface it serves, fins of one length hold 50.0% of the ceiling with their lumen free, and fins of unlimited lengths 83.3% ×1
with a reach cost of 2 every lining holds no more than it did with reach free, and more lengths still hold at least as much as fewer ×1
with a reach cost of 2 tube radii of channel per unit of reach, at k = 0.2 one length holds 50.0% of the ceiling and unlimited lengths 76.1%, against 50.0% and 83.3% when reach is free ×1
with channels sized in proportion to their flow, the branched supply and the vessel per fin both reproduce the reach model's supplied linings to a part in a million, and the wall integral agrees with a closed form to 1e-7 ×1
with no reach cost the lining solver reproduces the straight demand's crossings for three lengths, and with a reach cost it reproduces the one-length closed form, to two parts in ten thousand ×1
with reach free the ratio of one length to unlimited lengths follows the straight demand's closed form, 1 − 1 ⁄ (1 + 2k)² past a half ×1
with reach free, m lengths are supplied for nothing up to k = 1 ⁄ m(m + 1); with reach costing 2 the thresholds fall to 0.3333, 0.0938, 0.0435, 0.0250, 0.0162 ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

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.

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.

A shared supply is the same bill

A tube's fin lining charged for how far its supply has to carry was expected to do better under a supply that branches, one vessel running out along each fin and feeding everything it passes. It does not, and it cannot: with channels sized to the flow they carry, the per-unit charge and every branching tree are the same integral. Sharing pays only when a channel's width grows more slowly than its flow, as Murray's and Poiseuille's vessels do, and then branching buys back exactly what those economies would otherwise have charged, to within a fifth of a point.

Four finders, one option

Four unrelated lineages arriving at the same corrugation is read as evidence that the corrugation is good. It is at least as much evidence that there was nothing else to arrive at: how far a folded sheet shrinks is exactly its average layer count, so a lineage choosing a packing ratio is choosing a number of layers and nothing else — and the quantity that is genuinely free turns out to be almost uncorrelated with it.

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.

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.

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.

Nothing in a body folds on a line

A crease in an organism is not a crease. It is a compliant region — a patch of thinner material that bends — and a region has a width. The width consumes surface in exact proportion to the number of folds, which puts a ceiling on how fine a pattern can usefully get.

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.

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.

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.

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.

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.

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.

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.

Twice as thick where it is thickest

A folded leaf's thickness is quoted as an area calculation: so much lamina, so much footprint, so many layers on average. The average is not what has to fit in the bud. Sampling the folded state of corrugated leaf patterns of two to five rows gives a deepest point of eight, twelve, sixteen and twenty layers against averages of 4.15, 6.23, 8.31 and 10.39 — a ratio of 1.926 that does not move at all.

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.

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