Nothing grown has a seam

Nothing grown was cut out of anything: a leaf’s creases end where the plant put them, and the reason nothing in nature has a rim through the middle of its own structure is that nothing in nature was cut.

Nothing grown was joined either, and that turns out to matter in a different way.

The sheets a body has #

Biology folds surfaces to fit area into volume, which is the whole of why folded structures appear in bodies: a gut, an airway, a cortex, a mitochondrion’s inner membrane.

Those sheets come in two kinds.

Discs. A leaf, a petal, a wing case, a bract. A sheet with one boundary, grown outward from a meristem or a wing bud, with an edge that is where growth stopped.

Tubes. A gut, a blood vessel, an airway, a duct. Closed in one direction, with two open ends, grown as a tube from the beginning.

There is no third kind. Nothing in a body is a torus of membrane, nothing is a Möbius band, and nothing has a seam.

A grown tube and a joined one #

A gut and a paper tube are the same sheet: a cylinder, with two circles of boundary and one loop that cannot be shrunk.

Every condition this collection has established about a cylinder applies to both. A gut’s wall folded flat would have to cross an even number of creases round its circumference; the parity condition does not care how the sheet was made.

What differs is entirely in the history, and the history leaves no mark.

Where the difference shows #

Three places, and each is worth having because each is a place where a biological structure is unlike a manufactured one.

No seam. A joined tube has a line where the material meets itself — stiffer, thicker, a stress concentration, and the thing a designer measures from. A grown tube has nothing: every point of it is like every other, the wall is continuous, and there is no distinguished line anywhere.

No cut edge. A joined tube’s two ends are where the rectangle was cut. A grown tube’s ends are where it opens into something else — a mouth, a valve, a junction — and they are structures rather than terminations.

No period. A manufactured tube is a repeating pattern of a whole number of periods, because it was cut from a repeating drawing and had to match up. A grown tube’s pattern is continuous with itself by construction and has no period to be a whole number of.

Why a grown tube always closes #

That last point is the interesting one and it is the essay’s finding.

A manufactured tube can fail to close: cut a rectangle that is not a whole number of periods, join it, and the creases arrive at the seam at the wrong heights, meeting nothing. The construction refuses, and the reason is that the drawing had to match itself across a join that was imposed on it.

A grown tube has no join and nothing to match. Its wall’s structure is laid down on a surface that is already closed, so whatever pattern develops is automatically consistent with itself all the way round — not because anything checked, but because there was never a discontinuity for a mismatch to appear at.

So one whole category of failure is unavailable to biology, and the reason is developmental rather than mechanical.

The gut, in detail #

The clearest case is worth working through, since it is a tube with folds and it is the structure the surface-in-a-volume argument is usually about.

A gut is a tube, and its inner surface is folded to increase area: circular folds running round the tube, villi standing on those, microvilli on those. Three scales of folding, each multiplying the area.

Topologically the wall is a cylinder — closed round, open at both ends — and it is the same sheet a paper tube is.

The circular folds run round the tube, which is the axis that matters for the parity. A loop running once round the gut, at a fixed position along it, crosses no circular fold at all: it runs parallel to them. A loop running along the gut and back is contractible.

So the parity condition, which applies, has nothing to bite on: the creases run in the direction that does not cross the non-shrinkable loop.

That is a fortunate arrangement rather than a designed one, and it is the same arrangement a Miura tube has in one of its two directions. A structure whose folds run round a tube is a structure whose parity condition is vacuous.

Why a leaf is the easy case #

The other kind of biological sheet has no conditions at all beyond the local ones, and it is worth saying why.

A leaf is a disc: one boundary, no loop that cannot be shrunk, every closed path contractible. So the two-colouring is implied, the closure condition is automatic, and nothing global applies.

That is why the leaf’s folding has been so tractable in this collection — its patterns cost exactly one node of search per panel, its creases end where the plant put them, and nothing about it needs the machinery this phase built.

And it explains the asymmetry in what has been studied. The biological folding literature is mostly about leaves and insect wings, which are discs; the tubes are studied by physiologists asking about area rather than by anybody asking about folding.

The sheets nature does not make #

Two of the four sheets in this phase have no biological instance, and the absences are informative.

A torus of membrane. Closed in both directions, no boundary at all. Nothing in a body is one: every folded surface has an opening somewhere, because a surface with no opening encloses a volume that nothing can enter or leave.

That is a functional reason rather than a developmental one. A closed surface is a container, and a container that is folded to increase its area is increasing the area of something sealed — which is occasionally useful, as a vesicle, and vesicles are not folded.

A Möbius band. Non-orientable, one side. Nothing in a body is one, and here the reason is developmental: a tissue sheet has two faces that are biochemically different — an apical side and a basal side — and a surface where those two are the same surface is not a thing a polarised epithelium can be.

So biology’s sheets are exactly the ones with two sides and at least one opening, which is discs and tubes. That is a smaller set than the mathematics allows and the restriction has a cause in each direction.

What a two-sided tissue means #

The second absence deserves a moment, because it is a stronger statement than it looks.

An epithelium is polarised: its cells have a top and a bottom, the two faces do different things, and the difference is maintained by machinery inside every cell. A sheet of such cells has two sides in a chemical sense as well as a geometric one.

A non-orientable surface has no consistent side, so a polarised epithelium cannot form one. Walking round the surface, a cell would arrive back at its neighbour with its apical face against the neighbour’s basal one, and the tissue would have to resolve a contradiction that has no resolution.

That is a much firmer prohibition than nobody has observed one. It is a statement that the tissue’s own organisation forbids the topology, and it is the biological version of mountain and valley not being globally definable on such a sheet.

Two subjects, the same obstruction, and one of them expresses it as a crease assignment and the other as a protein gradient.

Growth as a way of avoiding a problem #

There is a general observation here about how development sidesteps a class of difficulty.

A manufactured closed object is made by taking a flat piece and joining it, and joining introduces a matching condition: the two edges have to agree. That condition can fail, and when it does the object cannot be made.

Growth never joins. A tube develops as a tube, from a sheet that was already curved into one, and its pattern is laid down on a surface with no discontinuity in it. So there is nothing to match and nothing to fail.

That is a real advantage of the developmental route and it is not the only one. The same holds for cutting: a grown structure has no cut edge, so it has no artefacts of where somebody cut, and every feature of its boundary is a feature.

Two operations that manufacturing needs and biology does not, and both of them are the source of a class of defect that biological structures simply do not have.

What the analogy is good for anyway #

Having listed the disanalogies, it is worth defending the comparison, since this collection makes it constantly.

The mathematics is about surfaces with creases, and a biological folded structure is one — approximately, for the purposes of the questions being asked. A gut’s wall does not stretch much on the timescale of a fold opening, and a leaf in a bud is close enough to inextensible that the packing arguments work.

What the analogy gives is the conditions: which foldings are possible on which sheets, what a loop costs, where a stack has a bottom. Those are exact statements about the idealised object and they are the right first approximation to the real one.

What it does not give is anything about growth, stretching, active motion or material. The organism is not the model is the collection’s standing caution, and it applies here with the usual force.

So the honest position is that a gut is a cylinder for the purposes of asking which folded states exist, and is not a cylinder of paper for any other purpose.

Which conditions still apply #

All the ones about the sheet, and it is worth being clear that the essay is not claiming biology escapes the mathematics.

The parity applies. A tube of any origin folds flat only if a loop round it crosses an even number of creases. A gut’s circular folds — the plicae — are creases running round the tube rather than along it, so a loop along the gut crosses them and a loop round it does not, and the condition falls on the wrong axis to bite.

The vertex conditions apply, unchanged, everywhere.

The closure condition applies: the composed motions round the tube have to give the identification, which for a grown tube is the identity map on a closed surface.

What biology does that paper cannot #

Two things, and both are why the analogy has limits.

It grows the pattern on the closed surface. Paper is flat and has to be joined; a tissue is laid down on whatever shape the embryo has. So biology gets closed sheets for free and paper pays for them with a seam.

It is not developable. A biological membrane stretches, thickens, grows differentially, and buckles into shapes no flat sheet could reach. The whole of this collection’s machinery assumes a surface that does not stretch, and a growing one does.

The second is the larger caveat and it applies to everything this collection says about biology. A gut is a cylinder in the topological sense and its wall is not paper.

What a body is optimising #

A closing note on why the sheets are the sheets, since the essay has been describing a restriction and restrictions usually have reasons.

The folded surfaces in a body exist to put area into volume: absorbing area in a gut, exchange area in a lung, computational area in a cortex. The quantity being maximised is area per unit volume, and folding is the mechanism.

For that purpose a tube is the obvious shape, because the thing whose area matters is a boundary between two compartments and a tube is a boundary with a compartment on each side.

And a disc is the obvious shape for a surface whose job is to face outward — a leaf, a wing — where the two sides do different things and neither is enclosed.

So the two sheets biology uses are the two shapes a functional surface has, and the ones it does not use are the ones with no functional interpretation. That is a satisfying explanation and it is a functional one rather than a developmental one, which means the two absences above have different causes: the torus is absent because it would be useless and the Möbius band because it would be impossible.

Three histories, one surface #

The essay’s structure is a comparison and it is worth setting the three out.

Cut from a larger sheet. A patch. Its boundary is where somebody cut, its features near the edge are truncated, and most of what a small patch is, is edge. This is what every figure in this collection shows.

Joined from a flat piece. A manufactured tube. Its boundary is two circles that were the cut ends, it has a seam where the join is, and its pattern had to match itself across the join.

Grown as itself. A biological tube. Its boundary is two openings that are structures rather than terminations, it has no seam, and its pattern never had to match anything.

Three histories, and the middle two produce the same surface. The mathematics sees the surface and none of the history, which is exactly right for the questions it asks and is why a claim about a cylinder covers both a gut and a straw.

What the history decides is which failures are available: a patch can have artefacts of where it was cut, a joined tube can fail to match, and a grown one can do neither.

A tube with no seam, in the hand #

The difference between a grown tube and a joined one is worth feeling, and there is a way to feel it without any biology.

Take a paper tube made by joining a strip, and run a finger round the inside. There is a step where the two edges meet: a change in thickness, a lip, a place the finger catches.

Now take a length of any seamless tube — a drinking straw, a cardboard tube wound rather than joined, a piece of pipe. Run a finger round it and there is nothing.

Both are cylinders. Every count in this collection is the same for both. What differs is a feature that no topological or combinatorial statement can see and that a finger finds in a second.

That is a useful calibration for how much the mathematics is throwing away. It describes the sheet exactly and says nothing about the seam, and the seam is what a person notices first and what a manufacturer worries about most.

The one condition a grown tube does carry #

To end on what does transfer, since the essay has been mostly about what does not.

A grown tube is a cylinder, and a cylinder has a loop that cannot be shrunk. If the tube’s wall is to fold flat — pressed, collapsed, packed — then the creases that loop crosses have to be even in number.

For a gut that is vacuous, because its folds run round the tube and the loop runs round it too. For a structure whose folds run along a tube it is not, and such structures exist: a collapsed blood vessel, an airway pressed shut, an insect’s tracheal tube.

Whether any of those exhibits the parity is not known here, and it is a checkable question: count the longitudinal folds in a collapsed vessel and see whether the number is even.

That is a small, concrete, testable prediction from a piece of folding mathematics about a biological structure, and it is the sort of thing this collection produces rarely enough to be worth flagging as one.

What is worth carrying #

A body’s folded surfaces are discs and tubes, they are grown as those shapes, and there is no seam anywhere.

The conditions this collection has established for glued sheets apply to the tubes, because the conditions are about the sheet. The failure modes of gluing do not, because nothing was glued.

And the difference between a grown tube and a joined one is invisible in the object and complete in its history — which is the same observation the leaf makes about cutting, arriving at the other end of the same distinction.