The most decided vertex here

A twist tessellation looks complicated. A patch of one holds dozens of polygons, each rotated against its neighbours, with pleats running between them in every direction and creases crossing the sheet at half a dozen different angles. Beside a box-pleating grid it looks like a different order of object.

At the level the conditions work at, it is the simplest thing here.

Sixteen, eight, four #

Four creases at a point can be lettered sixteen ways. Maekawa’s condition — the mountains and the valleys must differ by two — admits three-and-one or one-and-three, four arrangements of each, so eight.

The big-little-big lemma then says that the two creases bounding a sector strictly smaller than both its neighbours may not carry the same letter. Where it applies, it removes half of what is left: four.

Where it does not apply, the count stays at eight. And it applies much less often than one might expect, because the word doing the work is strictly: a vertex whose two smallest sectors are equal escapes it entirely, and so does a vertex whose sectors are all the same.

The vertices this collection folds are mostly of the escaping kind. A box-pleating grid’s are four right angles. A Miura’s, a leaf’s and a crumple’s are all degree four with no sector strictly smaller than both its neighbours in the arrangements that arise. Eight apiece, every one of them.

The twist corner #

Every interior vertex of a twist tessellation is the same object: a corner of a twist polygon, where two of the polygon’s own sides meet the two creases of the pleats running away from that corner.

Its four sectors are the polygon’s interior angle, the tiling’s own angle across the corner, and the two pleat sectors, and on the square tessellation at the proportions this collection draws they come out at

$$46.15°,\quad 90°,\quad 133.85°,\quad 90°.$$

The smallest is the pleat sector at $46.15°$, and both of its neighbours are right angles. It is therefore strictly smaller than both, the lemma applies, and the table falls from eight to four.

That is not a fact about the square tiling. On the triangular grid the four sectors are $60°$, $133.85°$, $120°$ and $46.15°$; on the honeycomb, $120°$, $133.85°$, $60°$ and $46.15°$. The pleat sector is $46.15°$ in every case — the same number, because it belongs to the pleat rather than to the tiling — and it is the smallest of the four every time.

The polygon’s interior angle at that corner is worth a sentence, because it is the one number in the list that comes from the tiling. Two sides of a twist polygon meet at $180°$ minus the tiling’s own angle between the two edges they face — a fact of the construction rather than a choice, since two lines whose normals differ by an angle meet at $180°$ minus it. On the square grid that gives $90°$; on the triangular grid, $120°$; on the honeycomb, $60°$.

So of the four sectors at a twist corner, two come from the tiling and two from the pleat, and it is always the pleat’s that is smallest. That is why the count is the same on tilings whose geometry is not.

Four, on every tiling #

The count is four on the square, the triangular, the honeycomb, the elongated triangular tiling and the rhombille. Every interior vertex of every patch, without exception.

That is a stronger uniformity than the tilings themselves have. The rhombille is the one whose vertices are not all alike — six rhombi meet at a lattice point and three at a triangle’s centre — and its twist polygons come in two sizes with two different side distances. Its crease pattern’s vertices are still all the same kind of object, because the construction puts a polygon corner at every one of them, and they all admit four.

The elongated triangular tiling has two kinds of tile and gives a patch with two kinds of corner — one with a $90°$ tile angle and one with $60°$ — and both admit four, because in both cases the smallest sector is the pleat’s and both of its neighbours are larger.

What four buys #

Half the cost per panel of anything else here.

The five patches cost 0.53, 0.47, 0.51, 0.52 and 0.51 search steps per panel. The grid, the leaf, the Miura and the crumples — the families with eight — cost exactly 1.00.

The mechanism is propagation. A table with four entries narrows to a single entry as soon as one of its creases is known, because the four entries are two pairs and knowing one crease eliminates half of them and then the lemma settles the rest. A table with eight entries narrows to two, which is still a choice, and a choice is a branch and a branch is a step.

So a patch’s letters are largely forced, and the search’s job on one is to read the forcing rather than to make decisions. Which is exactly the character the collection attributes to the families that fill their own sheet, and the patches have it more strongly than any of them.

The reputation, and where it came from #

The tessellation patches were the collection’s difficult family for a long time, and the reputation was earned by real measurements.

One patch’s search cost ran from eighty-six steps to past fifteen thousand depending on nothing but the seed. Three patches needed a quarter of a million steps to settle. Nine patches over a grid of construction parameters have no consistent lettering at all. Every one of those is true and none of them is about how constrained a vertex is.

The variance came from a coin in the search’s own arrangement and went away when the coin did. The expensive negatives are a different quantity from the cost of a positive, and proving a negative is dear for reasons that have nothing to do with the tables. The patches with no lettering are, if anything, further evidence for this essay’s reading: a highly constrained pattern is one whose conditions can more easily have no simultaneous solution.

What was never measured was the plain level: what a patch costs to letter, under a fixed order, per panel. It is half the line, and it was hidden under three other quantities that were all about something else.

What happens when the smallest sector changes hands #

The count is four because the pleat sector is the smallest, and the pleat sector is not always the smallest.

It grows as the turn closes. On the square tessellation it runs from $46.15°$ at a turn of $0.35$ radians up past $61.92°$ at $0.20$ and $75.21°$ at $0.10$ — always below the square’s own $90°$, so on that tiling the count never changes. On the triangular grid and the honeycomb the tile’s angle in the list is $60°$, and once the pleat sector passes it the smallest sector at a corner is a different sector.

That transition is where the collection found a patch going from having no lettering to having one across a hundredth of a radian, and it arrives here as a change in which condition applies where. Below the transition the lemma still applies — the tile’s $60°$ is strictly smaller than its neighbours too — but it constrains a different pair of creases, and the conditions that result have no simultaneous solution on those tilings.

So the count of four is stable and what it constrains is not, and the interesting behaviour of these patterns lives in the second rather than the first.

Constrained and rare are the same direction #

There is a pleasing consistency between this and the other things the collection knows about these patterns, and it is worth drawing out because it looks like a tension and is not.

A patch’s vertices are the most decided here, so a patch is cheap to letter. A patch’s letterings are also the rarest here: the share of drawn letterings that agree with themselves falls fastest on the tessellations, and nine of ninety-six patches over a parameter grid have none at all.

Both follow from the same constraint. Fewer admissible labellings per vertex means fewer admissible letterings of the sheet, which is rarity; and it means the propagation decides more, which is cheapness. A sampler throwing at a shrinking target does worse and a propagator walking a narrowing corridor does better, and the same number moves both.

The place where constraint stops helping is at zero. A pattern whose conditions have no simultaneous solution is cheap to refute and impossible to letter, which is what the nine patches are.

What this does not explain #

The constant, exactly. Four labellings a vertex gives about half a step per panel and eight gives one, and this essay offers no derivation of why half rather than a third or three-quarters. The relationship is measured across families and its magnitude is not accounted for anywhere here.

Nor does it say anything about a patch’s other difficulties, of which there are several. Whether the letters can actually be stacked is a separate question with no cheap answer. Whether the construction’s polygons overlap is a question about geometry. Whether a patch has any lettering at all is decided by the conditions jointly rather than by their count.

The claim is narrow: at the level of one vertex, a twist patch is the most decided pattern here, and that is what its cost per panel follows.

The number that does not change with the tiling #

There is one more thing in the sector list worth pulling out, because it is the reason the count is the same across five tilings that share almost nothing.

The two pleat sectors are $46.15°$ and $133.85°$ on the square, on the triangular, on the honeycomb and on the elongated tiling alike — the same two numbers, to two decimals, on tilings whose tiles are squares, triangles, hexagons and mixtures. They are set by the turn and the fill of the construction and by nothing else.

The two tiling sectors are not: $90°$ and $90°$ on the square, $120°$ and $60°$ on the triangular, $60°$ and $120°$ on the honeycomb.

So the four sectors at a twist corner are two universal numbers and two local ones, and the count of four depends on the universal pair being smaller than the local pair. On the tilings and at the proportions this collection draws, they are. That is a condition rather than a theorem, and it is the condition that fails when the turn closes far enough — which is where the family’s interesting behaviour comes from.

What a designer takes from it #

A twist tessellation is a good pattern to letter by machine and a bad one to letter by eye, and both halves come from the same number.

By machine: four labellings a vertex means the letters are nearly forced, so a correct lettering is available for any patch a designer is likely to draw, cheaply, at any size.

By eye: four labellings a vertex means that a wrong letter is caught quickly at the vertex where it is wrong — the local conditions really do bite — and it does not mean the sheet is right. A patch whose every vertex checks out can still have a ring of relations that closes on itself, and the construction’s own suggested lettering is exactly such a case: it satisfies every condition at every vertex and forces a loop of twenty-eight panels.

So the constraint that makes a machine’s job easy does not transfer to a person’s, because the thing a person can check is the part the constraint has already handled.

Where else the lemma applies #

Four is unusual here and it is not unique, and the other places it turns up are worth naming because they say what kind of vertex the lemma actually catches.

It catches a Miura vertex drawn at an angle that makes one sector strictly smallest, which some Miuras are and the ones measured here are not. It catches the vertex of a twist unit drawn on its own, which is the same corner as a tessellation’s. And it catches any vertex whose four sectors are four different sizes, since among four distinct numbers arranged in a ring at least one is strictly smaller than both its neighbours.

What it misses is every vertex with a tie among its small sectors, and ties are what regular constructions produce. A grid has four equal sectors. A preliminary base has two pairs. A Yoshimura at its natural proportion has six equal. Regularity is exactly what makes a tie likely, and a tie is exactly what silences the lemma.

So the rule of thumb runs against intuition: the more symmetric a vertex looks, the less the conditions decide about it. A twist corner is asymmetric in a way that costs nothing to draw — its four angles are simply four different numbers — and that asymmetry is worth a factor of two in what its conditions settle.

Which theorem was checked, and how #

The tables are enumerated rather than derived. All sixteen labellings of a degree-four vertex are generated, filtered by Maekawa’s difference and by the lemma applied at each sector in turn, and what remains is counted. Four is a count.

The sectors are measured off the drawn pattern: the creases at each vertex are found by incidence, sorted by the angle at which they leave, and the sectors are the differences between consecutive angles. So a patch drawn differently from the description above would give different angles and a different count, and the count is checked against the drawing rather than assumed from it.

And the uniformity is checked across every interior vertex of every patch rather than at a representative one. A single vertex admitting eight would show as a range rather than a single number, and on all five tilings the range is a point.

A count that was always available #

The last thing worth saying about this number is that nothing had to be built to find it.

The tables have been computed at every vertex of every pattern in this collection since the first search was written; they are what the search propagates over. Four has been the answer for the tessellation patches for as long as there have been tessellation patches, sitting inside every run, never printed.

What was missing was the comparison — the same quantity read off every other family and set beside it. A number that is only ever computed on one kind of object is a number with nothing to be large or small against, and it took a census over nine families to notice that four is the smallest here and that the smallest is the cheapest.

That is a cheap kind of finding and this collection should probably look for more of them. The instruments produce a great deal that is used and not reported, and a quantity used at every step of every search is exactly the kind that becomes invisible.

The angle the count rests on #

Everything above turns on one sector being smaller than the other three, and that sector is worth measuring rather than describing.

It is the pleat’s, and it belongs to the pleat: $46.15°$ at a turn of 0.35 radians and a fill of 0.62, on the square tiling, the triangular grid, the honeycomb and the elongated tiling alike, and at every pitch from a fifth of a sheet to a half. The tiling does not enter it, because a pleat is a strip of fixed proportional width and the angle its two creases leave a corner at is a local matter between one polygon’s side and its neighbour’s.

Which is why the count of four is the same on tilings whose geometry is not: two of the four sectors at a twist corner come from the tiling and two from the pleat, and the pleat’s are the small ones.

What the picture cannot show #

A drawing of a patch does not show which of its sectors is smallest, and that is the entire content of the essay. At the scale a whole patch is drawn, a $46.15°$ wedge and a $60°$ wedge are the same wedge, and the difference between them is what decides whether a condition applies at every vertex of the sheet.

Nor does a table size show which labellings survive. Four is a count of arrangements, and which four they are — and therefore what the propagation does with them — is a fact about the vertex that no number records.

And a count taken at every vertex of a patch is a count over the patch’s interior vertices only. The ones the rim leaves short of paper are asked nothing at all, and there are a great many of them — most of a patch is edge — so the uniformity reported here is a uniformity among the vertices the pattern actually constrains, and the ones outside it are not four or eight but nothing.