Origami & Folding
what one sheet can do
Flat-folding

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.

Assumes Most of a patch is edge and Cutting a patch out of a plane.

20 min read 8 figures The pattern is the objectOne sheet, no cuts

Everything folded in this collection is a disc of paper. It has an edge, the edge is where the paper stops, and every piece of machinery assumes so: a folded state is built by walking outward from a starting panel until the panels run out, a vertex is interior when it has a full turn of paper around it, and a stacking is a list of panels from the bottom one to the top one.

A tessellation is not a disc of paper. It is a rule for filling the plane, and the squares this collection has been drawing of it are specimens. Reading a specimen and reporting what one finds is a perfectly good way to work — most of a patch is edge has been a useful thing to know, and so has what the edge does to the counts — but it leaves a question that no amount of drawing squares can settle: which of the things a square says are about the tessellation, and which are about the square.

Answering it needs the tessellation itself, as an object something can be run on. This essay is how one is built.

The rectangle that repeats

Every tiling here has a rectangle it repeats in. On the square grid it is one tiling unit by one. On the triangular grid, the honeycomb and the rhombille it is one by 3\sqrt3; on the elongated triangular tiling it is one by 2+32 + \sqrt3.

None of those is the smallest cell the tiling has. The three middle ones are generated by steps of (1,0)(1, 0) and (12,32)(\tfrac12, \tfrac{\sqrt3}{2}), whose smallest cell is a rhombus, and every clip, face walk and figure in this collection takes a rectangle. The rectangle 1×31 \times \sqrt3 is a cell of the lattice generated by (1,0)(1,0) and (0,3)(0,\sqrt3), which sits inside the full one at index two: it is twice as much paper and exactly as periodic. That costs a factor of two in the panel count and buys everything else, which is a good trade.

Multiply the rectangle by a whole number in each direction and it still repeats. So a cell can be one period, or four, or nine, and the same construction describes all of them.

One period of the square twist tessellation, with its edges joinedThe crease pattern of a single repeating cell of a twist tessellation on the square grid, drawn on the rectangle it repeats in. The rings mark where a crease meets a side of the cell: each one on the left is the same crease as one on the right, and each on the bottom the same as one on the top. Joined that way the 12 pieces are 8 creases, the 9 drawn panels are 4, and all 4 vertices are interior.one period of the square grid's twist tessellationa ring is where a crease leaves and returns on the far side12 crease pieces → 8 creases9 drawn panels → 4 panels4 vertices, every one interiorV − E + F = 0mountainvalleyraw edge
Fig. 1 The smallest cell of the square twist tessellation: four panels, four vertices, eight creases. Every ring is a crease leaving one side and returning at the other.

Where to put the corner

A rectangle has to be placed somewhere, and one placement is much better than the others.

The joining works by matching: a crease meeting the left side at a given height is the crease meeting the right side at the same height, and a panel touching one side is the panel touching the other. Matching by position is only unambiguous when the sides do not run through anything interesting. A side passing exactly through a twist polygon’s corner would have several creases arriving at one point and no way to say which is which; a side passing through a vertex would cut a vertex in half and leave two objects that are neither interior nor boundary.

So the corner goes where the sides are furthest from every drawn point. Reduced into one period, the x coordinates of every crease end form a ring of values; the largest gap in that ring is where the vertical sides should fall, and the same argument in the other coordinate places the horizontal ones. Two one-dimensional questions rather than one two-dimensional one, and it is the stricter reading as well as the faster: a point is a hazard if it is near a vertical side or a horizontal one, so both coordinates want clearing independently.

A cell whose best placement still leaves a side within a ten-thousandth of a sheet of some drawn point is refused rather than joined, because an identification matching creases that arrive at the same place is an identification matching the wrong things.

On the tilings here the clearance that placement achieves is comfortable — a few hundredths of a sheet width, against creases whose length is measured in tenths — so the refusal has never fired on a cell that was wanted. It exists because the failure it prevents is silent: two creases matched to each other by accident produce a perfectly well-formed object whose every count is plausible and whose relationship to the tessellation is nothing at all.

What the joining does

Three identifications, and each is a different kind of object being glued.

Creases. A crease running off one side runs back on at the other, so the two pieces the drawing shows are one crease and must carry one letter. On the two-period square cell that is forty drawn pieces and thirty-two creases: eight pairs.

Panels. A panel touching the left side and the panel touching the right side at the same height are one panel of the sheet, cut in two by the rectangle. Twenty-five drawn panels become sixteen. Each joining also records a displacement — the left-hand half is the right-hand half moved a period over — and those displacements are what make the pattern’s layer relations readable later.

Vertices. Nothing at all, and that is the interesting one. The sides are placed to miss every vertex, so a point where a crease crosses a side has two creases in line with each other and is not a vertex. The pattern’s vertices are exactly the ones inside the rectangle, and every one of them is now interior — not because anything was done to them, but because there is nowhere left to be otherwise.

What joining the edges does to the countsOne row per glued cell: how many panels the drawing shows and how many the sheet has, how many crease pieces are drawn and how many creases those are, how many vertices there are, and Euler's number. Every one of the 15 cells gives V − E + F = 0, which is what a torus gives.gluing a cell's opposite edges, on five tilingspiecespanelsdrawncreasesverticesV−E+Fsquare ×19412840square ×225164032160square ×349368472360triangular ×123123424120triangular ×2694811696480triangular ×31391082462161080hexagonal ×123123424120hexagonal ×2694811696480hexagonal ×31391082462161080elongated ×133205240200elongated ×210580184160800elongated ×32171803963601800rhombille ×137246048240rhombille ×212196216192960rhombille ×32532164684322160a torus has V − E + F = 0, and these three counts are made three different ways
Fig. 2 Fifteen cells over five tilings, before and after joining. The last column is the check.

The check that could fail

Three separate acts of identification produce three counts, and Euler’s formula relates them: a torus has V−E+F=0V - E + F = 0.

Sixteen vertices, thirty-two creases, sixteen panels. Twelve, twenty-four, twelve on the triangular cell. Twenty, forty, twenty on the elongated. Every cell here gives nothing, over five tilings at three sizes.

That is a test rather than a restatement, and it is worth being clear about why. The vertex count is read off the crease graph by a routine that decides whether each vertex has a full turn of paper around it. The crease count is the result of a union-find over crease pieces paired by their endpoints on the sides. The panel count is the result of a different union-find, over faces found by walking the planarised graph, paired by probing across the sides at each face’s rim edge. Three procedures with almost nothing in common, and a mismatch in any of them moves one count without moving the others.

There is one more reason to like Euler’s formula as the check. It is a global statement, and the errors most likely in this construction are local — a single unmatched crease at one corner, a single panel joined to the wrong partner. A local error moves exactly one of the three counts by a small amount, and the sum is the only place a small amount is visible, because sixteen panels and seventeen panels look equally reasonable written down on their own.

What joining the edges does to the countsOne row per glued cell: how many panels the drawing shows and how many the sheet has, how many crease pieces are drawn and how many creases those are, how many vertices there are, and Euler's number. Every one of the 9 cells gives V − E + F = 0, which is what a torus gives.gluing a cell's opposite edges, on five tilingspiecespanelsdrawncreasesverticesV−E+Ftriangular ×123123424120triangular ×2694811696480triangular ×31391082462161080hexagonal ×123123424120hexagonal ×2694811696480hexagonal ×31391082462161080rhombille ×137246048240rhombille ×212196216192960rhombille ×32532164684322160a torus has V − E + F = 0, and these three counts are made three different ways
Fig. 3 The three tilings built on the triangular lattice, whose repeating rectangle is one unit by √3. The counts differ and the last column does not.

The three other checks

Euler’s formula is the one that catches a systematic error. Three more catch inconsistencies.

Two routes must agree. Panels get joined in a chain — this one to that one across the left side, that one to a third across the bottom — and a panel reached two ways must be recorded as being the same number of periods away either way. The disagreements are counted and they are zero, with the underlying arithmetic agreeing to about one part in a hundred million million.

Every partner must exist. A crease ending on a side without a match on the opposite side, or a panel touching a side with nothing across from it, is counted. Both counts are zero on every cell here. A single orphan would mean the rectangle was not a period.

Every relation must belong to one crease. The folded state produces a relation for each pair of panels sharing a fold, and each has to be matched to exactly one crease of the drawing. A relation matching two creases, or none, is counted. Zero again.

One period of the triangular twist tessellation, with its edges joinedThe crease pattern of a single repeating cell of a twist tessellation on the triangular grid, drawn on the rectangle it repeats in. The rings mark where a crease meets a side of the cell: each one on the left is the same crease as one on the right, and each on the bottom the same as one on the top. Joined that way the 34 pieces are 24 creases, the 23 drawn panels are 12, and all 12 vertices are interior.one period of the triangular grid's twist tessellationa ring is where a crease leaves and returns on the far side34 crease pieces → 24 creases23 drawn panels → 12 panels12 vertices, every one interiorV − E + F = 0mountainvalleyraw edge
Fig. 4 The triangular tiling’s smallest cell: twelve panels, twelve vertices, twenty-four creases in a rectangle one unit across and √3 up.

None of these four is expensive, and all four run on every cell the collection builds. That is deliberate: a construction that is checked once when it is written and never again is a construction whose next parameter change breaks it quietly, and the failures worth fearing here are the silent ones.

What the object is good for

Three things, each of which was previously a guess.

It makes a controlled comparison possible. The same rectangle, the same drawing, the same vertices — cut out or joined — with the rim as the only thing that varies. That comparison inverts what the rim was thought to be doing, and no arrangement of patches could have made it.

It exposes a hypothesis. The test this collection uses for whether a lettering describes a possible stack of paper turns out to be a test for a disc, and on a sheet with no edge it is wrong. Nothing in the code says so, because nothing outside the class had ever been built.

And it gives the folded sheet’s own lattice, which is a quantity nothing here could previously compute. A panel and its partner across a side are one panel of the sheet, so the difference between the motions the fold gives them is where a cell of paper goes — and that turns out to be one similarity for every tiling.

What cutting a sheet out of a tessellation addsEach bar counts the creases that a rectangular cut divides, which become two independently lettered creases on the cut sheet and are one crease on the glued one. The note gives the two crease counts and the number of vertices, which is the same either way: the cut runs between the vertices and changes no condition asked of any of them.what a cut adds, in letterssquare ×148 creases become 12 · 4 vertices either waysquare ×2832 creases become 40 · 16 vertices either waysquare ×31272 creases become 84 · 36 vertices either waytriangular ×11024 creases become 34 · 12 vertices either waytriangular ×22096 creases become 116 · 48 vertices either waytriangular ×330216 creases become 246 · 108 vertices either wayhexagonal ×11024 creases become 34 · 12 vertices either wayhexagonal ×22096 creases become 116 · 48 vertices either wayhexagonal ×330216 creases become 246 · 108 vertices either wayelongated ×11240 creases become 52 · 20 vertices either wayelongated ×224160 creases become 184 · 80 vertices either wayelongated ×336360 creases become 396 · 180 vertices either waythe bar is how many creases the cut divides; nothing else about the two sheets differs
Fig. 5 The size of the difference between a square of a tessellation and the tessellation: the creases the square divides, which are two letters on the square and one on the pattern.

The counts a square gets wrong

Set the two objects side by side and the specimen’s distortions are easy to name.

A two-period square of the square tessellation reports twenty-five panels; the pattern has sixteen. Nine of the twenty-five are halves and quarters of panels the square divided, and any statement of the form this pattern has so many panels per unit area read off the square is wrong by more than half.

It reports forty creases; the pattern has thirty-two. Eight of the forty are halves of creases, and each pair is reported as two independent objects with two independent letters, which is exactly the freedom that makes a square of a tessellation so much cheaper to letter than the tessellation.

It reports sixteen vertices, and that one is right, because the square was placed to miss them.

So the distortion is not uniform: it is severe in the panel count, severe in the crease count, and absent in the vertex count. A reader taking any per-unit-area quantity off a patch is taking a number whose error depends on which quantity it is, and the errors do not cancel.

What a period is not

Two things are worth guarding against, because both look like the object described here and are not.

It is not a bigger patch. A patch is a disc with a rim; nine periods of drawing cut into a square is still a disc with a rim, and it has the same slack a one-period square has, only proportionally less of it. Enlarging a patch approaches the tessellation in some senses and never reaches it, because the rim never goes away.

It is not a pattern with periodic boundary conditions in the folder’s sense. Nobody folds this. There is no sheet of paper whose left edge is its right edge, and the object is not a proposal for one. It is the pattern, considered as the mathematical thing every claim about a tessellation has always been about, with the finite square removed from the description.

Five tilings, five cells

The construction is the same on all five and the objects it produces are not, in a way that is worth reading as a fact about the tilings.

The square gives the smallest cell: four panels, four vertices, eight creases at one period. Its repeating rectangle is one tiling unit square, and its twist polygons are squares, so a period holds exactly one twist.

The triangular grid and the honeycomb give the same counts as each other — twelve panels, twelve vertices, twenty-four creases — which is not a coincidence, since the honeycomb’s vertices are the triangular grid’s tile centres and both repeat in the same 1×31 \times \sqrt3 rectangle. Their patterns are different and their cells are the same size.

The elongated triangular tiling’s rectangle is 1×(2+3)1 \times (2 + \sqrt3), nearly four times as tall, and its cell holds twenty panels — because a period of that tiling contains a row of squares and a row of triangles rather than one kind of tile.

The rhombille gives twenty-four panels, twenty-four vertices and forty-eight creases, the largest of the five, because it is the one tiling here whose vertices are not all alike and a period has to contain one of each kind.

So cell size tracks how much of the tiling a period has to hold, which is a statement about the tiling’s symmetry rather than about the twist construction laid over it. Every one of the five satisfies the same four checks.

What the reader can still fold

Everything, and by the ordinary route.

A period is a description, so any patch of the tessellation is a piece of it and prints exactly as it always did. What the joined object adds is that a lettering found on it can be written back onto any patch of any size, and the letters agree wherever two patches overlap, because they came from a rule about the plane rather than from a search on one square. That is the practical difference: a periodic lettering is one answer for every patch at once, and a lettering found on a patch is an answer for that patch.

The lettering that was proved impossible, checked on paper with an edgeEach bar is one clipped patch carrying the periodic lettering, its length the number of creases. Every patch passes all four vertex conditions and has no forced loop in its layer order, on 3 tilings and at 3 sizes.the impossible lettering, on ordinary patchessquare ×140 creases16 vertices · every condition holds · no forced loopsquare ×2144 creases64 vertices · every condition holds · no forced loopsquare ×3312 creases144 vertices · every condition holds · no forced looptriangular ×1116 creases48 vertices · every condition holds · no forced looptriangular ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×1116 creases48 vertices · every condition holds · no forced loophexagonal ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×3924 creases432 vertices · every condition holds · no forced loopthe bar is the crease count; the note is what the ordinary checks said
Fig. 6 One periodic lettering, written onto nine ordinary patches at three sizes. The same rule fits all of them.

The cost of building one

Modest, and worth stating because the construction is not free.

The drawing has to be generated over enough of the plane to cover the cell with a ring of complete twist polygons around it, because a polygon whose neighbour has none is drawn with a side and without one of its pleats — an odd vertex, failing the angle condition, several tiling units away from anything the cell contains. That failure shows up as a cell whose panels will not close, at a distance from its cause, and it is the one mistake in this construction that took real work to find.

So a polygon missing a neighbour is not drawn at all, and a cell asked for beyond the region that was generated is refused rather than clipped. The refusal is the important half: a cell with creases quietly missing at one corner would pass every check in this essay.

One period of the hexagonal twist tessellation, with its edges joinedThe crease pattern of a single repeating cell of a twist tessellation on the honeycomb, drawn on the rectangle it repeats in. The rings mark where a crease meets a side of the cell: each one on the left is the same crease as one on the right, and each on the bottom the same as one on the top. Joined that way the 116 pieces are 96 creases, the 69 drawn panels are 48, and all 48 vertices are interior.one period of the honeycomb's twist tessellationa ring is where a crease leaves and returns on the far side116 crease pieces → 96 creases69 drawn panels → 48 panels48 vertices, every one interiorV − E + F = 0mountainvalleyraw edge
Fig. 7 Two periods of the honeycomb’s twist tessellation, which is the largest cell drawn in this essay: forty-eight panels, forty-eight vertices, ninety-six creases.

What the picture cannot show

A cell is drawn as a rectangle with a boundary, because that is the only way to draw it, and the boundary in the picture is exactly the thing the object does not have. The rings marking where creases cross it are the closest a static drawing gets to saying so.

Nor can a picture show that the pattern continues. Every figure here shows one period and the reader has to supply the plane, which is the same act of imagination any tiling figure asks for and slightly more consequential here, because in this case the continuation is where the mathematics lives rather than being decoration around a specimen.

And a rectangle is a choice. The pattern repeats under a whole lattice of translations, and the rectangle is one convenient cell of one convenient sublattice of it — twice the smallest cell on three of these tilings, and no more canonical than any other. Nothing in the object depends on the choice; every count above scales with it, and the checks hold at one period, four and nine alike. But a reader who wants the smallest description of a triangular twist tessellation will not find it here, and the reason is that a rhombus is harder to clip to than a rectangle rather than anything about paper.

One period of the rhombille twist tessellation, with its edges joinedThe crease pattern of a single repeating cell of a twist tessellation on the rhombille tiling, drawn on the rectangle it repeats in. The rings mark where a crease meets a side of the cell: each one on the left is the same crease as one on the right, and each on the bottom the same as one on the top. Joined that way the 60 pieces are 48 creases, the 37 drawn panels are 24, and all 24 vertices are interior.one period of the rhombille tiling's twist tessellationa ring is where a crease leaves and returns on the far side60 crease pieces → 48 creases37 drawn panels → 24 panels24 vertices, every one interiorV − E + F = 0mountainvalleyraw edge
Fig. 8 The rhombille tessellation’s smallest cell, which is the largest of the five: twenty-four panels, twenty-four vertices, forty-eight creases, because the rhombille has two kinds of vertex and its repeating rectangle holds more of the tiling than the others do.

Where this goes next

the same tessellation on the same square, cut out of the plane two waysassembled from whole unitsclipped from the plane12 crossings · panels 1.73 apart0 crossings · panels closemountainvalleyraw edge Cutting a patch out of a plane A tessellation is infinite and a sheet is not, so every picture of one is a decision about where the paper stops. Assembling whole twist units on a square and running the outstanding pleats to the rim puts 12, 18, 12 and 5 creases across other creases on four of five tilings; generating the pattern over a larger region and clipping it puts none. The panels then place exactly — and what is waiting behind the repair is a different refusal that could not be asked about before. the bar is the share of the twists on the paper that the paper's edge cuts0.5 of the sheet86%1 whole · 6 cut by the edge0.42 of the sheet55%5 whole · 6 cut by the edge0.34 of the sheet59%7 whole · 10 cut by the edge0.28 of the sheet70%7 whole · 16 cut by the edge0.22 of the sheet37%17 whole · 10 cut by the edge0.18 of the sheet49%23 whole · 22 cut by the edgea patch is a picture of a tessellation, and the smaller the unit the less of the picture is edge Most of a patch is edge Between 34% and 91% of the vertices in the crease patterns drawn here sit on the edge of the paper rather than inside it, and on the tessellation patches — the figures that are meant to show what a repeating pattern looks like — it never falls below a third. A boundary is one unit deep whatever the unit is, so the share falls like one over the number of units across and reaches nothing at any size a page can carry. the same drawing, cut out of the plane and glued upnodes, log scale, against periods across the sheet10100100010⁴10⁵1×12×23×34×4glued upcut outan open mark is a search that ran out of budget rather than a cost 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.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 11 that link here.

The objects this essay names

Each one links to every other essay that touches it.

BoundaryBoundary vertexCrease patternInterior vertexPanelPeriodicitySymmetryTessellationTilingTwist