Periodicity — where it appears
Named by 41 essays across 8 fields — each of them below, with the objects they name alongside it.
One vertex, repeated
Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.
A square that turns
A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.
Which polygons twist
Twist tessellations come in three kinds — triangle, square, hexagon — and it is natural to read that as a fact about twists. It is not. A twist can be built around any regular polygon and every one of them folds; what stops at three is the tiling, and the tiling is a fact about the plane.
A unit that folds is not a tessellation
Of the 512 repeating rules for the waterbomb tessellation, 56 pass every condition on a two-by-two patch and 32 pass on every larger one. The twenty-four that die were never foldable — the small patch simply contained one of the four kinds of vertex the pattern makes, and the failures were at the other three.
Letters that agree get rarer
Two hundred letterings drawn independently from a square twist tessellation patch, and twenty-six of them have letters that do not contradict themselves. On the next patch up it is five, then two, then none, then none. What the share falls with is not the size of the patch and not the angle of its twist: it is the number of independent closed chains its panels form, which is Euler's relation on the drawing and is fixed before a single letter is chosen.
A corrugation agrees with itself
A Miura fold of forty-eight panels and a twist tessellation patch of forty-nine have almost exactly the same number of independent closed chains for their letters to contradict themselves round — thirty-five against thirty-six. Sixty-four per cent of the Miura's drawn letterings are consistent and thirteen per cent of the patch's. A Yoshimura at thirty-three chains manages ninety-three. The room to fail sets the scale; the construction decides where in it a pattern lands.
Sixty-four rules, sixteen fold
The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.
What the rim was doing
One rectangle of a twist tessellation, cut out of the plane in the ordinary way, gives up a consistent lettering in forty-eight steps. Join its opposite edges so that no crease is divided and the same drawing, at the same vertices, under the same conditions, takes fifty-six thousand seven hundred and seventy-two. The edge of the paper was never the difficulty. It was the slack.
A loop that goes somewhere
Every crease says which of its two panels lies above the other, and a loop in those statements is a proof that the pattern has no flat folded state. On a sheet with no edge that sentence is false. The loops of a periodic pattern carry a lattice step each, and a loop that ends one cell to the right is not a contradiction — it is a stack of paper with no bottom layer.
The lettering that was proved impossible
A search closed its whole tree on a glued square tessellation and reported that no mountain-and-valley assignment of it is consistent. Written onto ordinary patches of one, four and nine periods and handed to the four vertex theorems and a folded sheet rebuilt from scratch, the assignment it says cannot exist passes every check, on four tilings, up to fifteen hundred creases.
A sheet with no edge
A twist tessellation repeats, so a rectangle of it is a description of the whole plane rather than a piece of paper. Joining the rectangle's opposite sides makes that explicit and produces an object every gate in this collection can read: twenty-five drawn panels become sixteen, forty crease pieces become thirty-two, sixteen vertices are all interior, and the three counts add to nothing.
The rim is four letters a cell
Cut a rectangle out of a tessellation and it asks exactly the vertices the tessellation asks, exactly the same questions. What it adds is four free letters for every period of edge — the creases the cut divides, which become two independently answerable creases instead of one. Eight letters on a two-period square, sixteen on a four-period one, and nothing else about the two objects differs at all.
The bottom layer is at the rim
A hundred and sixty-nine panels of folded tessellation, and three of them have nothing underneath. All three touch the paper's edge, and the same is true on every tiling at every size measured. Which panel is at the bottom of a stack turns out to be a fact about where the sheet was cut rather than about the pattern, and the pattern itself has no bottom at all.
Pruning on proofs alone
A search that discards a branch it cannot prove wrong is not a search. Deciding whether a periodic pattern's layer relations really contradict themselves is far dearer than the disc's one-pass test, so the cheap test is asked first — it is sufficient, so it settles almost everything — and the expensive one runs only on what the cheap one rejects. Five of nine steps on a small cell, fifty thousand of fifty-seven on a large one.
The cost of proving something false
A search closing its whole tree is the strongest result this collection can produce, and on a glued tessellation it produces one that is wrong. What it costs to reach is three steps at one period, thirty-five at four, three thousand four hundred and fifty-five at nine, and more than two hundred thousand at sixteen — growing far faster than the cost of finding the lettering it says does not exist.
Where you cut hardly matters
Slide the same rectangle across one whole period of the same tessellation and every position gives a different patch: different creases divided, different half-panels round the edge, panel counts from forty-nine to sixty-one. The cost of lettering them runs from twenty-five steps to thirty-three. Whether a cut is made changes the answer by three orders of magnitude; where it falls changes it by a third.
Folding it flat is one similarity
Where a cell of paper goes when a twist tessellation collapses is a scale and a turn: multiply the plane's lattice by 0.410373441 and rotate it by 36.62°. That is the answer on the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the rhombille alike, agreeing to eight figures — while the collection's other answer to how much smaller it gets gives those five tilings five different numbers.
The edge was not what made it hard
Five families of pattern searched at one step per panel and a tessellation patch did not, and the property left standing after four alternatives were killed was having a rim. Measured under a fixed letter order the patches cost between a half and two-thirds of a step per panel, at every tiling and every size — below the line rather than above it, and the rim is why.
A count is not a length
Cut a rectangle out of a tessellation and it reports fifty per cent more creases than the pattern has, then twenty-five, then seventeen — converging on the truth from above and never reaching it. The crease length per unit area it reports is exact at every size, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not.
An order with no least element
Enumerating every way a folded pattern can be stacked works by building upward from a panel with nothing below it. The smallest square twist patch has exactly one such stacking and takes eleven thousand steps to find it. The pattern that patch was cut from has no panel with nothing below it at all, so the enumeration has nothing to start from — and the sheet is perfectly well stacked anyway.
A test imported without its hypothesis
The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.
Half a rim
A rectangle of tessellation cut out of the plane has four edges; glued into a torus it has none. Gluing one pair and leaving the other gives the middle of the scale — the same drawing, the same vertices, the same conditions asked of them, and exactly half the rim. What the rim costs turns out to be measurable per edge rather than only at the ends.
The rim adds up
What one glued pair of a cell's edges saves in free letters is what the other pair saves, and gluing both saves the sum. That is a rate rather than an observation, it is the form of the claim two objects could never support, and it is what makes 'the rim costs four letters a cell' a statement about tessellations rather than about one drawing.
Euler counts the gluing
Vertices minus creases plus panels comes to one on a rectangle of paper and nought on any gluing of it. That is the cheapest check that an identification did what it says, it costs three counts already being made, and it is what found a crease running exactly through the corner of a cell — a case the corner search could not see and no other check would have noticed.
A grid that will not close
Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.
The corrugation that closes on itself
A Miura cell crosses one crease per period in one direction and four in the other. So one of its two directions has a parity condition that half the sizes fail, and the other direction cannot fail at any size — the same sheet, the same drawing, and two gluings that behave completely differently.
One node per panel, with the rim gone
A rectangle of repeating pattern cut out of the plane costs exactly one node of search per panel, on every family and at every size. Take the rim away and the total falls and the cost per panel rises, because the letters that were removed were the ones that could not be wrong.
The turn a column costs
The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.
The period nobody measured
Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.
Which pair is glued
A cell's two cylinders have the same Euler number, the same amount of rim and the same name. On a symmetric drawing they have identical counts of letters, panels and vertices — and searching them costs twenty-four nodes one way and eighty-five the other. Half the rim is a description of the topology and not of the object.
A tessellation on a cylinder
A twist tessellation has been drawn here as a patch and as a torus, and never as anything in between. Gluing one pair of a cell's edges gives the family its first sheet with exactly two edges — the shape every folded tube actually has, and the only object in the collection that can say whether the rim's cost is linear in how much rim there is.
The seam that is not a symmetry
Gluing a cell's edges looks like a symmetry of the drawing and is not. It is an instruction about which points of the paper are the same point, the drawing has to agree with it along the whole of a glued edge, and a rectangle that is not a period of the pattern does not glue at all — which turns out to be the only real restriction on which cylinders exist.
A metamaterial with no edge
A folded metamaterial's properties are quoted per unit cell, because a material is supposed to be the same everywhere and a cell is supposed to stand for the whole of it. Every cell this collection has measured has been cut out of a patch, with a rim round it — and a rim is the one place a repeating material is not like itself.
A bottom layer on half a rim
The bottom of a folded stack lives at the paper's edge, which is why a sheet with no edge has an order with no least element. A cylinder has half a rim, so it has a bottom — and the count of panels that could be it falls with the rim, which makes the claim a measurement rather than a boundary case.
The drawing does not say what is glued
One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.
The symmetry a gluing adds
A patch of a tessellation has whatever symmetry its outline allows — a few reflections, a rotation or two. Glue its edges and it acquires translations, and a lettering of the glued sheet has to be invariant under them. That is a much stronger requirement than a lettering of the patch, and it is why one answer covers every patch at once.
Nothing grown has a seam
A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.
The helix chooses the lattice
A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.
A domain too short to be unique
A staple holds the scaffold by pairing with a stretch of it, and two arguments decide how long that stretch has to be. One is combinatorics — a stretch of seven bases has about four hundred other places in a 7,249-base strand it would also match. The other is thermodynamics, and it is the one that binds: a duplex that is unique at eleven base pairs still comes apart at the temperature the design is held at, and staying paired takes seventeen. Rounded up to the crossover period, that is three periods on both lattices, and the lattice the helix prefers is the one whose three-period domain leaves some of its staples unattached.
The plane the five points were in
Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.
Each drawing has its own threshold
Gluing a cell's edges was measured once, at one size, and found to cost three orders of magnitude — which cannot tell a threshold from a slope, nor say whether a cut sheet has one further out. Swept from one period to five on four tilings, every sheet starts at about a third of a node per free letter and every drawing leaves that behaviour at a size of its own: four periods on the square grid, three on the honeycomb, two on the triangular grid and two on the rhombille, where even the cut sheet crosses.
Named alongside it
The objects these essays reach for when they reach for this one.
BoundaryTessellationPanelGluingAssignmentPatchUnit cellCrease assignmentConstraintInterior vertexLayer orderSearch cost