Surface area — where it appears
Named by 9 essays across one field — each of them below, with the objects they name alongside 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.
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 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.
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.
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.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Surface in a volumeTrade-offScalingConstraintOptimisationEfficiencyIdealisationThicknessCorrugationPacking efficiencyPacking ratio