A tree cannot argue

A molecule is what fills a polygon of a crease pattern once the polygon’s shape has been decided: creases running from the outline inward, arranged so that the paper collapses onto a single axis. The last free parameter is the account of how much freedom remains once the polygon is fixed, and the answer is a little — the construction is mostly forced.

Where the creases come from is the straight skeleton: shrink the polygon inward at a uniform rate and trace where its corners go. The skeleton changes its mind at the events where two of those traces meet, and the traces themselves become the creases.

A straight skeleton of a simple polygon is a tree. It has no closed circuit in it: shrink a polygon and the corner traces meet and merge, they never come back round to where they started. That is a fact about the construction and it turns out to decide something the construction was not designed for.

What a pattern’s letters need in order to go wrong #

A crease pattern’s letters can contradict themselves. Each crease says which of the two panels it joins lies above the other, and a chain of panels each of which must lie below the next is a proof that no order exists and so no flat folded state.

Such a chain is a closed chain of panels, and how many independent ones a pattern has is arithmetic: edges less nodes plus pieces on the graph whose nodes are panels and whose edges are creases. On every pattern this collection draws that number comes out equal — or nearly — to the count of interior vertices.

Maekawa closes every chain that goes round a single vertex. So what is left is combinations of them, and the number of combinations grows as two to the power of the count.

A fold-and-cut pattern has one to three. A tessellation patch has thirty-six to a hundred and twenty-six.

Two hundred and eighty letterings, none of them wrong #

The measurement is straightforward. Draw forty independent letterings from each of the seven outlines this collection folds and cuts — a triangle, a square, a rectangle, a pentagon, a house, an L and a five-pointed star — and ask each whether its letters agree.

All two hundred and eighty do.

The star is the extreme case and the clearest. Its skeleton collapses to one node — no perpendicular survives, every foot landing off the end of its edge — so the pattern is ten creases through the centre and the panel graph has exactly one chain in it. There is nothing for a contradiction to sit on but the one thing that cannot carry one.

What the count is, precisely #

The number being quoted is the panel graph’s cycle rank: edges less nodes plus connected pieces, on the graph whose nodes are the pattern’s panels and whose edges are its creases. Boundary edges are not creases and contribute nothing, which is why a fold-and-cut pattern’s panel graph comes out in two pieces rather than one — some panels touch the rest only across the sheet’s own edge.

That is worth stating because the count is otherwise easy to compute wrongly. Counting all edges including the boundary gives a much larger number and a meaningless one: a boundary edge joins a panel to the outside, and the layer relation has nothing to say about the outside.

Computed correctly, the seven outlines come out at one, one, two, one, three, three and one. The largest of them has fifteen panels and three chains, which is fewer chains than a single square twist unit.

Why a shrink cannot come back #

The claim that a straight skeleton is a tree is the load-bearing one and it deserves more than an assertion.

Shrink a simple polygon inward at a uniform rate. Every edge moves parallel to itself; every corner traces a line along the bisector of its two edges. Two things can happen: two corner traces meet, in which case the edge between them has vanished and the two traces merge into one; or a corner trace reaches an edge on the far side, in which case the region divides.

Both events reduce the number of live traces or split the region, and neither creates one. So the traces form a structure that only ever merges as the shrink proceeds, and a structure that only merges is a forest — a tree per connected region. There is no operation in the process that could bring a trace back to a place it has already been.

That is why the skeleton has no circuit, and it is why the crease pattern built from it has no circuit either: the creases are the traces, plus the perpendiculars dropped from the nodes, and a perpendicular runs from a node to an edge without joining two nodes.

Why this is not quite a theorem #

It would be neat to say that a molecule can never have a contradictory lettering, and that claim is stronger than the evidence.

What is established is that a skeleton has no circuit of creases, so the pattern’s chains are the vertex chains and nothing else generates. Maekawa closes each generator. What is not established is that no combination of two or three of them can be oriented — a chain enclosing two adjacent skeleton nodes is a legitimate closed chain of panels, and nothing local forbids it.

The house and the L have three chains apiece and therefore seven non-trivial combinations each. Forty draws of each found none closed. That is a measurement, not a proof, and it is reported as one.

What can be said with confidence is the comparative claim, and it is the useful one. A construction producing one to three chains is in a different regime from one producing a hundred and twenty-six, and the regime is what decides whether a lettering has to be checked at all.

The “or nearly” is removable #

The relation between chains and interior vertices was stated above as an approximate one, and it is exact, which matters because the exactness is what turns “the vertex chains generate” from an observation into an identity.

The panel graph is the crease pattern’s dual with the outer face and the boundary edges removed. In a plane graph, the independent cycles of that dual are in one-to-one correspondence with the vertices it encloses — a cycle of panels is a loop drawn in the plane, and what a loop in the plane can enclose is a set of interior vertices. So the cycle rank equals the number of interior vertices, and the seven outlines bear it out exactly: one, one, two, one, three, three and one, against interior-vertex counts of one, one, two, one, three, three and one.

The chain space is therefore spanned by the single-vertex chains, for any crease pattern whatever, not merely for skeletons. What the tree property does is guarantee there is nothing else — no chain that fails to enclose a vertex, which is what a circuit in the crease graph would supply.

Why spanning is not enough #

That is also the precise reason the argument stops short of a theorem, and it is worth stating in the terms that make the gap visible rather than as a shrug.

The chain space is a vector space over the two-element field: chains add, adding a chain to itself gives nothing, and a spanning set generates every chain by addition. Maekawa says each generator is not a contradiction.

But “is not a contradiction” is a statement about orientations, and orientations do not add. A chain is contradictory when its layer relations run the same way all the way round; two chains can each fail to close while their sum closes, because the sum’s orientation on a shared edge is not determined by the two summands’ orientations on it. Acyclicity is not a linear condition, so a spanning set of acyclic generators does not span an acyclic space.

That is the whole of the gap, and it is the reason the house and the L needed measuring rather than deducing. Three generators give seven non-trivial sums, and each of the seven is an independent chance for an orientation to close that neither of its parts could.

It also says where a proof would have to come from: not from counting chains, which is settled, but from the geometry of which layer relations a skeleton’s creases can carry.

What the chain count is, on these #

The numbers are small enough to state one by one, and doing so makes the structural claim concrete.

The triangle, the square, the pentagon and the star each have one interior vertex and one chain. The rectangle has two. The house and the L have three. Nothing here has four, and the largest of them is fifteen panels.

Compare a square twist unit at four chains, a Miura of six by four at fifteen, a Yoshimura at twenty-two, and the smallest tessellation patch at thirty-six. The fold-and-cut outlines are not merely at the low end of the range; they are below the smallest designed pattern in the collection.

Two other constructions with the same property #

The molecule is not the only radial construction here, and the two others behave the same way, which is what makes the account structural rather than a fact about molecules.

The fold-and-cut construction is the same skeleton put to a different purpose: instead of collapsing a polygon onto an axis it folds an outline onto a single line so that one straight cut releases it. Same tree, same absence of circuits, same result — all seven outlines consistent in all forty draws.

A single vertex is the degenerate case and it is worth including. A cone of creases meeting at one point is a tree with one node, one chain, and that chain is the one Maekawa closes. Every admissible labelling of it is consistent, at every degree, and that is a theorem rather than a measurement.

So the family is: constructions whose creases radiate from a set of points and never return. Everything in it is safe on this question, and the reason is the same in each case.

What that means for designing with molecules #

Circle packing and the tree method build a base by assigning each flap a circle, packing the circles, and filling the polygons between them with molecules. The output is a crease pattern whose letters have to come from somewhere, and this says where the risk is not.

The molecules themselves are safe, in the sense that the letters inside one polygon have almost nowhere to disagree. What is not covered by any of this is what happens between them: the creases joining one molecule to the next, and the rivers between packed circles, which are where a pattern built this way has whatever circuits it has.

That is the honest reading, and it is a narrower guarantee than a designer would want. It says a molecule is not the part to worry about — not that a base is safe.

It is also worth naming what a molecule’s letters are usually decided by, since the risk being low does not mean the choice is free. The construction assigns them: the creases from the skeleton’s nodes to the polygon’s corners take one letter and the perpendiculars the other, because that is what collapses the region onto the axis. So a molecule arrives lettered, and the letterings measured above are alternatives nobody would draw. The measurement is about how much slack the construction has, not about a decision anyone faces.

What a guarantee of this shape is worth #

There is a habit worth naming here, because this collection has been on the wrong side of it.

A construction that has never produced a bad output is not the same as a construction that cannot. The tessellation builder had never produced a visible failure either, for several rounds of work, and it turned out to have been shipping a patch whose letters forced a chain of twenty-eight panels the whole time — the failure was there and nothing was reading the crease list in the way that would have found it.

What separates the molecules from that case is not a better record. It is that a reason can be given: the skeleton is a tree, so the chains are the vertex chains, so a contradiction has almost nowhere to sit. A reason of that kind survives the construction being changed, extended to new outlines, or run at a size nobody has tried, and a clean record does not.

The reason here is partial — it explains why there is little room and does not prove there is none — and a partial reason with a measurement beside it is what this is. That is a better position than either alone.

The contrast, stated once #

Put the two constructions side by side and the difference is structural rather than a matter of care.

A molecule is built from a shrinking process, and a shrinking process produces a tree. Its creases radiate; nothing comes back.

A twist is built from a polygon that rotates, and the polygon is a closed ring of creases. Its whole defining feature is a circuit, and the circuit is where its letters fail.

Neither construction was chosen for this property and neither designer was thinking about layer relations. It falls out of what each construction is for: a molecule is a way of collapsing a region onto an axis, which is a radial operation, and a tessellation is a way of repeating a unit, which is a periodic one. Radial constructions produce trees and periodic ones produce grids.

The exception a hole would make #

There is one shape of outline this argument does not cover, and it is worth marking because the collection can draw it.

A polygon with a hole — an annulus, or a shape with an island — has a skeleton that is not a tree. Shrinking it inward from both boundaries produces traces from the outer edge meeting traces from the inner one, and the meeting places form a closed circuit going round the hole. That is a genuine circuit of creases, and a pattern built on it has chains that no vertex accounts for.

A hole is an edge and a ring and a line are the two places this collection has met the same structure from other directions: a sheet with a hole has closed chains of panels going round the hole, and on an odd number of creases it has no two-colouring at all.

Nothing here folds and cuts a holed outline, so there is no measurement. The prediction is clear enough to record: a molecule filling a holed polygon should behave like a small twist rather than like the outlines above.

What the reflex corner did not change #

The corner that splits the shrink is the case where the skeleton’s behaviour is least obvious: a reflex corner runs outward as the polygon shrinks, eventually meets an edge on the far side, and the shrinking region divides in two. That event was refused by this collection’s construction for a long time and is now solved for.

It does not disturb any of this. A skeleton with split events in it is still a tree — the division makes a node with more arcs coming out of it, not a circuit — and the two non-convex outlines that the split events made reachable, the L and the star, are both in the two hundred and eighty above.

The star is in fact the pattern with the fewest chains of any this collection draws. An outline that turns back on itself five times, whose skeleton refused to be computed at all until recently, produces the crease pattern with the least room for its letters to go wrong.

That is worth holding on to as a caution about intuition. The outlines that look difficult — reflex corners, spiky stars, shapes the construction refused — produce the tamest crease patterns here, and the pattern that produces contradictory letterings nine times in ten is a square twist repeated nine times on a square grid.