Two surfaces in one box

The surface has to be supplied charges the channel that services a folded surface, and finds it charged on exactly the same terms as the sheet’s own thickness. The correction is a factor and the factor is large, and the shape of the answer does not change: one surface, one box, one parabola.

A body does not have one surface. It has a gut lining and a vascular bed and a lymphatic network and sheets of muscle, and they are in the same volume doing different things — none of them cut out of anything or joined at a seam. The obvious way to model that is to give each of them a share of the depth, and the obvious expectation is that the shares add back up to what one surface would have had.

They do not. They add up to a fraction of it, and the fraction is one over the number of sharers.

The ceiling is quadratic in the depth #

The arithmetic is in the ladder’s first rung and is worth extracting, because everything here follows from it.

A corrugation of k folds in a box of side S with depth D available holds k·S·(D − k·t) of surface, and that peaks at k = D ⁄ 2t with a value of S·D² ⁄ 4t. The corrugation is the uniform one, because at a stated ratio there is no other.

The optimum fold count is linear in the depth. The area it reaches is quadratic in it. Halving the depth halves the number of folds worth putting in and quarters the area they hold.

That is not obvious and it is not an artefact. It happens because the depth does two things at once: it sets how many folds fit, and it sets how deep each panel can be. Halving it halves both, and the area is the product.

Which makes sharing expensive #

Now let m surfaces divide the box. Give each one a depth of S ⁄ m — a fair split, and the most favourable one, since an unequal split is worse by the same convexity that makes the split cost anything at all.

Each surface reaches S(S⁄m)² ⁄ 4t = S³ ⁄ 4tm². Multiply by m for the total:

total = S³ ⁄ 4tm

against S³ ⁄ 4t for a single surface with the whole depth. The total is the single-surface ceiling divided by m, exactly.

So the intuition — that shares add back up — fails by a factor of m, and the failure has a clean cause. Each sharer is penalised twice: once for having less depth, and once again for having fewer folds worth putting in it. The first penalty would be recovered by adding the shares back up; the second would not, and it is the one the quadratic carries.

What that means for a body with jobs to do #

The result is stated as a cost and it is really a constraint on architecture, so it is worth turning round.

A volume that has to perform several exchange functions can arrange them two ways. It can interleave — one folded sheet doing every job at once, with different regions of it specialised — or it can segregate, giving each function its own sheet with its own share of the depth.

Interleaving pays nothing. Segregating pays a factor of m in total area, and the factor is not asymptotic or approximate: it is exact for every m, at every box size and every sheet thickness.

That is a strong argument for interleaving, and bodies do interleave a great deal — a villus carries its own capillary and its own lymphatic inside itself rather than beside itself, which is the supply charged per fold rather than per share, and the epithelium, the basement membrane and the vessel are stacked within one fold rather than occupying three folded systems.

It is also an argument that says exactly what segregating buys, which is nothing that appears in this model. A segregated system can have different fold counts, different materials and different failure modes, and those are the reasons to pay a factor of m. The model prices the cost and is silent about the benefit, which is the right division of labour for a geometric argument and is worth stating so that the result is not read as a claim that segregation is a mistake.

The two corrections compared #

This ladder now has two corrections to the single-surface curve and they behave differently enough to be worth setting side by side.

Supply enters the coefficient. The charge per fold goes from t to t + δ, the optimum and the peak both fall by t ⁄ (t + δ), and the correction is a factor that a bigger box does not help with.

Sharing enters the depth. The available depth goes from S to S ⁄ m, the optimum falls by m and the peak by m², and the total across the sharers falls by m.

The second is the steeper of the two by a long way. A supply channel three times the sheet’s own thickness costs a factor of four. Splitting a box four ways costs a factor of four in the total — and a factor of sixteen in what each surface individually reaches, which is the number that matters if one of the four has to do a job on its own.

And the two compound in the obvious way, because they act on different parts of the same expression. m surfaces each of thickness t serviced at depth δ hold, in total, S³ ⁄ 4m(t + δ). A body with four segregated systems and a supply three times the sheet’s thickness reaches a sixteenth of what one unserviced sheet with the whole box would.

An unequal split is worse, and by how much #

The factor of m is computed for an equal division and an equal division is the best case, which is worth demonstrating rather than asserting because the direction is the one that strengthens the result.

Give two surfaces depths D₁ and D₂ with D₁ + D₂ = S. Each reaches S·Dᵢ² ⁄ 4t, so the total is proportional to D₁² + D₂². Subject to a fixed sum, a sum of squares is smallest when the parts are equal and grows as they separate: an eighty-twenty split gives 0.64 + 0.04 = 0.68 of the whole-box value against 0.5 for the even one.

Which reverses the reading. An unequal split is better for the total, not worse — the convexity runs the other way from the one the equal-split case suggests, because the objective is a sum of squares rather than an average of them.

That is worth pausing on, because it is easy to get backwards and the consequence is real. The best way to divide a box between two surfaces, if total area is what is wanted, is to give one of them nearly all of it. The even split is the worst two-way division, and it is the one the factor of m is computed at.

So the factor of m is a lower bound on what an even split costs and an upper bound on what any split costs, and the two-way case runs from a factor of two at even shares to a factor of one at a split so lopsided that one surface has vanished. A body that segregates and then gives one system almost all the depth has hardly paid anything; a body that segregates evenly has paid the full factor.

Which suggests the arrangement to look for. If total exchange area were the objective, segregated systems would be markedly unequal in the depth they occupy. They are — a gut wall is mostly epithelium and lamina propria with the muscle layers thin against them — and this arithmetic is one reason among several why that would be so.

What counts as a sharer #

The factor is the number of sharers, so the result is only as sharp as the question of what a sharer is, and the model gives a precise answer that is narrower than the word suggests.

A sharer is a system that needs its own depth, exclusive of everything else, across the whole extent of the box. Two systems that alternate through the same depth are one sharer with two kinds of layer. Two that lie against one another and exchange across the contact are one sharer with two faces. Two that occupy different parts of the box rather than different parts of its depth are two boxes, not two sharers.

That excludes most of what a naive count would include. An epithelium and its basement membrane are not two sharers; they are one sheet with a laminated cross-section, and their combined thickness is the t of a single surface. An epithelium and the capillary bed serving it are not two sharers either — the bed lies within the fold, so it is the δ of the rung below this one and enters the coefficient.

What does count is a system with a different topology through the volume. A lymphatic network that runs where the vascular one does not; a sheet of smooth muscle that has to be continuous around the tube rather than following the folds; a nerve plexus at its own level. Those need their own depth and cannot be folded into somebody else’s.

So the practical m for a real organ is small — two or three rather than ten — and the factor is correspondingly modest. The result’s force is not that bodies pay a large factor; it is that the factor exists at all, and that it is not the zero a fair-shares intuition predicts.

Why nesting is the arrangement that escapes this #

There is an architecture that beats every split and it is the one bodies actually use, so the essay owes a statement of why it escapes the arithmetic rather than merely doing better within it.

Splitting divides the depth: each surface gets S ⁄ m and folds within it. Nesting subdivides the fold: a corrugation whose panels are themselves corrugated at a finer scale uses the same depth twice, because the fine folding happens inside the coarse folding’s panels rather than beside them.

The two are not variants of one arrangement. In a split, the depths add to S. In a nest, they multiply — the coarse level buys a factor and the fine level buys another factor of the area the coarse one produced. A gut lining is folded at the scale of a plica, again at the scale of a villus, and again at the scale of a microvillus, and the three factors multiply.

That is why the factor of m is a real constraint and not a description of anything. The arrangement that pays it is the one bodies avoid, and this rung’s value is in saying what avoiding it is worth rather than in describing an architecture.

The arithmetic of the nested case is not in this rung and it is not a small extension: each level of nesting has its own sheet thickness, its own optimum and its own ceiling, and the levels interact because a fine fold inside a coarse panel is limited by the panel’s own depth. Working that out is the next rung, and the reason it is next rather than here is that it needs a recursion where this rung needs a division.

Which theorem was checked and how #

The quadratic is checked by the arithmetic it implies rather than by being written down. The figure requires the product of each sharer’s peak and its own m to reproduce the single-surface ceiling — which is the statement that the total is one m-th of it — and computes each peak by searching over whole fold counts rather than evaluating a formula.

The single-surface case is required to be in the comparison. A figure drawing only splits would have no baseline to be a fraction of, and the generator refuses a list of sharer counts that leaves it out.

The optima are whole numbers and the closed forms are not, so the check is stated to within half a fold. That tolerance is a property of counting folds rather than a slackening of the claim, and a wrong exponent breaks it by far more than half a fold at the sharer counts drawn.

And every quantity here is in units of the box. No organ is measured, no tissue thickness is quoted, and the sharer counts are chosen to span the range a body plausibly covers rather than to describe any body.

Where the model stops #

The split is equal and a body’s is not. An unequal split is worse than an equal one, by convexity — the sum of squares of parts summing to a constant is smallest when the parts are equal — so the factor of m is the best case and a real division does worse. That is the direction that strengthens the result and it is worth saying which way it runs.

The sharers are independent and stacked. Two surfaces that lie against one another and share a boundary are not two surfaces in this sense; they are one surface with two faces, and the model has no way to charge them correctly. Real tissue is full of that arrangement, which is another reason the interleaved case is the common one.

The depth is the only shared resource. A real volume also shares blood, mechanical support and space for the things the surfaces are exchanging with, and each of those is its own budget with its own arithmetic.

And nothing here forbids a body from being far from any optimum. The ceilings are ceilings. A structure at a third of its ceiling with four sharers may be doing better than one at its ceiling with none, if the four are doing four things — and the organism is not the model in any case.

What the picture cannot show #

The curves are what a box would hold and not what any body holds, and there is nothing on the page that distinguishes a structure sitting at its peak from one sitting well below it.

The figure also cannot show the arrangement that makes the whole comparison moot. Interleaving is not m sharers with a bigger m or a smaller one; it is a different architecture in which the count does not apply, and it appears in this ladder only as the m = 1 curve — which is to say the model’s way of drawing “one surface doing every job” is the same line as its way of drawing “one surface doing one job”.

That collapse is the honest limit of the argument. The model can price segregation and cannot represent integration except as its absence, so it establishes that segregating costs a factor of m and says nothing whatever about what integrating costs, which in a real tissue is certainly not nothing.

The idealisation, named #

The box is a fixed depth and the sharers are slabs within it, each with its own corrugation and no interaction between them. Three things are being assumed and each is worth naming.

The surfaces do not nest, and the folded state’s own depth is where a nested one would have to go. A corrugation inside another corrugation’s folds would use depth twice, and the model has no way to express it. Real structures nest constantly, at every scale from a villus to a microvillus, and each level of nesting multiplies the area rather than dividing the depth — a leaf’s corrugation does the same thing once — which is the arrangement that beats everything here and which this ladder has not priced.

The depth is partitioned rather than shared in time. A structure that folds one system flat while another is open uses the same depth twice, and nothing here charges for that.

And the sheet’s own thickness is the same for all the sharers. Give each sharer its own thickness and the total becomes a sum of S³ ⁄ 4tᵢm² terms, which is smaller than the equal-thickness answer whenever the thicknesses differ — so heterogeneity is a further cost on top of the split.

Where the ladder goes next #

The nesting case is the one this ladder now owes, and it is the interesting one because it is the arrangement that escapes the arithmetic above.

A corrugation whose panels are themselves corrugated has an area that multiplies rather than divides. Each level of folding takes a share of the depth and buys a factor, and the factors compound — so the question is how many levels a given depth supports and what each costs, which is a recursion with the same parabola inside it at every step. That is a computation this collection can make and has not made, and it is where the surface-in-a-volume field goes when it stops treating the box as something to be partitioned and starts treating it as something to be subdivided.

Sideways from here, the ceiling’s quadratic in the depth is the same shape the convergence ladder’s hinge bound has in the surface, and reading the two together makes the general form visible: a fold buys a linear benefit and costs a linear amount of some budget, and the ceiling is therefore quadratic in the budget. Any resource a fold consumes behaves this way, which is why halving any of them is so much more expensive than it sounds.

The thing worth carrying is the sentence that makes the factor of m predictable rather than surprising. When a ceiling is quadratic in a resource, dividing the resource m ways divides the total by m rather than leaving it alone. Fair shares are only fair when the thing being shared enters linearly, and it usually does not.