Origami & Folding
what one sheet can do
A pattern to fold

The hexagon twist

Six pleats arranged round a hexagon, and a good deal harder on the hands than it looks. Where the smaller unit collapses in one motion, this one wants all six pleats persuaded upward together: close them one at a time and the last has nowhere left to go, so the usual advice is to pinch opposite pairs and let the middle find its own rotation. Thirteen panels, eighteen creases, and 916 mm of folding on a sheet 150 mm across — a third again as much travelling as its smaller relative asks for, which is felt rather than read. The corner sectors arrive in two sizes, 120° and 60°, because six sides meeting a ring cannot make them all alike. They still come in equal pairs, so nothing is strictly smallest and the lemma about a smallest sector stays silent here as well; the letters have to be hunted for. Of 262,144 ways of lettering it, 4,096 survive every condition, and those separate into sixty-four families no amount of pushing on a pleat moves between — one family per way of lettering the ring itself. Handedness is settled at the drawing board, in other words, and a folder who wanted the middle to turn the other way needs a different sheet rather than a different technique. Only three regular polygons fill the plane, so only three of these units repeat; this is the roomiest of them, and the one whose uncreased middle is largest relative to its paper. That middle is worth a moment: it takes no crease at all, and on a folded specimen it survives as a flat lid sitting proud of everything around it, which is why photographs of these things tend to be photographs of that lid. Two practical notes for anybody printing it. Paper choice matters more here than on the smaller unit, because the folded sheet averages 3.3 layers over its footprint against the square's 3.0 and the ring gathers the deepest of them in one place: thin stock will take that without bruising, and heavier cartridge will fight it and then crack along the innermost line. And the order to work in is outside in — set every pleat as a shallow valley first, let the whole ring rotate a few degrees by itself, and only then press anything home. Reversing that order produces a stubborn, slightly starred object which will not flatten and cannot be rescued without unfolding the lot. What is not claimed here is anything about a sheet covered in these. One of them, drawn on its own square, has been put past every condition the subject has; a plane of them repeating is a separate question with a separate and much worse answer, and the essays are careful to keep the two apart.
The hexagon twistA hexagon twist: a hexagon with a pleat running out from each of its 6 corners, drawn at 150 mm and carrying 9 mountain and 9 valley creases — 916 mm of folding on a sheet 150 mm across. As the sheet closes the hexagon rotates, which is what gives the family its name. The sector angles are fixed by Kawasaki and the assignment is chosen for having a folded state rather than for reading well — and the unit is verified, while the tessellation it belongs to is not.6 corners, all alikesectors 120°, 120°, 60°, 60°two equal pairs, so no sectoris strictly the smallestthe assignment18 creases, searched9 mountain, 9 valleythe ring takes two lettersthe panels can be orderedwhat is checked6 interior verticesand not the tilinga twist of radius 0.17 sheet-widths18 creases, 6.11 sheet-widths of foldingmountainvalleyraw edge
hexagon twist — sheet 150×150 mm — 9 mountain, 9 valley, 916.32 mm of crease

Fold it

The sheet is on the printed page, at the size it says.

Printing this page gives the page, and adds one more: the pattern alone, at 150 mm across, on a sheet of its own. Cut a square that size, transfer or trace the lines, and fold. The size is repeated in the corner of the sheet, because a printer set to fit to page rescales silently and there has to be some way to notice.

Mountain and valley are distinguished by dash as well as by colour, so the sheet survives the monochrome laser printer it will almost certainly come out of.

The hexagon twist — sheet 150×150 mm — 9 mountain, 9 valley, 916.32 mm of crease

What it is

ProvenanceGenerated here, from Kawasaki's condition
Creases9 mountain, 9 valley
Interior vertices6, every one checked
Panels 13, read off the pattern
Folding to doabout 916 mm of crease at this size
Printed sheet150 mm across

What was checked

Four theorems at every interior vertex, and the faces two ways.

  • Developability — the sectors around each of the 6 interior vertices sum to a full turn, so the sheet was flat before it was creased.
  • Kawasaki — alternating sectors sum to a straight angle at each of them.
  • Maekawa — mountains and valleys differ by exactly two.
  • Big-little-big — no strictly smallest sector is flanked by two creases of the same letter.
  • The faces — 13 of them, found by walking the planarised graph, and checked against Euler's formula and against the area they cover. A face walk that goes the wrong way round or merges two faces usually still satisfies Euler; it does not conserve area.

None of this decides whether the whole sheet folds flat, which is NP-hard in general. Every local condition holds. That is a different and weaker statement, and it is the one being made.

Take it away

The field's own interchange format, so the pattern is reusable outside this site.

hexagon-twist.fold — 22 vertices, 34 edges, 13 faces, 5437 bytes. It opens in ORIPA, Rabbit Ear and the rest of the FOLD ecosystem.

The export is short because this repository never converts anything: FOLD's vertices_coords, edges_vertices and edges_assignment have been the in-memory representation of a pattern here since the site's first phase. What the file adds is the metadata that makes it openable, and the faces where they can be read. Coordinates are in sheet widths, and the file says what one unit measures on paper.

What is argued with it

Essays that call twist-unit — read off the figure index rather than listed by hand.

nearly flatwidth ×0.91 length ×0.98ν = -0.22half closedwidth ×0.66 length ×0.88ν = -0.52nearly packedwidth ×0.45 length ×0.55ν = -2.93both dimensions shrink together — pulling it open in one direction opens it in the other

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

tessellation 4 corners, all alikesectors 90°, 90°, 90°, 90°two equal pairs, so no sectoris strictly the smallestthe assignment256 of 4096 fold6 mountain, 6 valleythe ring takes two lettersthe panels can be orderedwhat is checked4 interior verticesand not the tilinga twist of radius 0.17 sheet-widths12 creases, 4.70 sheet-widths of foldingmountainvalleyraw edge

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.

tessellation polygonsectors at the twist vertexKawasakitiles the plane3-gon60.0 · 60.0 · 120.0 · 120.0180.0° = 180.0°yes — 6 round a point4-gon90.0 · 90.0 · 90.0 · 90.0180.0° = 180.0°yes — 4 round a point5-gon108.0 · 108.0 · 72.0 · 72.0180.0° = 180.0°no6-gon120.0 · 120.0 · 60.0 · 60.0180.0° = 180.0°yes — 3 round a point7-gon128.6 · 128.6 · 51.4 · 51.4180.0° = 180.0°noevery one of these twists satisfies the local theorems and folds flat on its ownthe interior angle has to divide 360° for the twists to meet, which only 3, 4 and 6 dobeyond 7 sides the assignment search runs out — 21 free creases, and the enumerator refuses above 22

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.

tessellation two kinds of vertex, both forced16 of degree 490°, 90°, 90°, 90°9 of degree 690°, 45°, 45°, 90°, 45°, 45°40 mountain and 36 valley creases14.3 sheet-widths of foldingmountainvalleyraw edge

The base that tiles

The waterbomb base is the first thing most people fold and the last thing they think about. Repeat it across a sheet and it becomes a tessellation with two kinds of vertex, an assignment that has to be searched for rather than remembered, and a folded state thirty-two times smaller than the paper.

tessellation what the construction produced9 twists, 36 interior verticesturned 24.1° from the tiling's edgespleats 0.118 to 0.118 wide2.20× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge

Any tiling makes a twist

A twist tessellation is usually drawn, admired and copied. It can be derived instead: hand the construction any tiling of the plane and it returns a crease pattern that folds flat, with the twist polygons' shapes forced by the tiling's own angles and nothing left to choose but how large and how turned.

tessellation patternclassdatedfoldingThe preliminary basetraditional—724 mmThe Miura foldpublished as mathematics19701049 mmThe square twistgenerated here—704 mmThe hexagon twistgenerated here—916 mmThe Yoshimura patternpublished as mathematics19552380 mmFold and cut — the trianglegenerated here—258 mmThe tapered corrugationgenerated here—1057 mmThe waterbomb tessellationtraditional—2290 mm2 dated, all of them published; 6 undated, none of them ownedthe fourth class — a designer's model — is what this shelf holds none of

The patterns nobody owns

This site prints crease patterns at true scale and prints no designer's work, and that has always been stated as a rule applied at the end. Read the printed shelf as a documentary record instead and the rule turns out to be a property of the record: every pattern that carries a date was published as mathematics, every undated one belongs to nobody, and the two silences are one silence.

history a single vertex is always one piece; a pattern with more is notand the number of pieces is decided by the creases that never reach the edge of the paperThe preliminary base1 vertices inside the paper112 letterings admitted1 piece of 1120 creases buried2^0 = 1The square twist4 vertices inside the paper256 letterings admitted16 pieces of 164 creases buried2^4 = 16The hexagon twist6 vertices inside the paper4096 letterings admitted64 pieces of 646 creases buried2^6 = 64Fold and cut — the triangle1 vertices inside the paper30 letterings admitted1 piece of 300 creases buried2^0 = 1

The creases that cannot move

One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.

flat-folding uncut: 4 vertices inside the paper, 6.3% of letterings admittedcut a crease with an interior vertex at each end25.0% admitted2 vertexes released · 4× the share · 4 such creases, all alikecut a crease that already reaches the edge12.5% admitted1 vertex released · 2× the share · 8 such creases, all alikea released vertex is one the four conditions no longer reach, and each is worth a factor of two

A cut is a licence

What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.

material The preliminary base: 8 symmetries, 112 letteringsthe bar is the share of letterings the symmetry carries to themselvesa quarter turnnone of 112 — this symmetry cannot be foldeda half turnnone of 112 — this symmetry cannot be foldedthree quarters of a turnnone of 112 — this symmetry cannot be foldeda mirror across the sheet12 of 112a mirror up the sheet12 of 112a mirror in one diagonal12 of 112a mirror in the other diagonal12 of 112

The symmetry the letters cannot keep

Every pattern in this subject is drawn symmetric and the symmetry is always quoted of the drawing. A folded object is a drawing and a lettering together, so a symmetry survives only if the letters keep it — and the preliminary base loses every rotation while the square twist, drawn with the same eight, loses the other half.

design The square twist, one crease at a timethe bar is the number of pieces, and a cut anywhere reduces itbefore any cut16 pieces · 0 vertices releasedcutting a crease that reaches the edge4 pieces · 1 vertices releasedcutting a buried crease2 pieces · 2 vertices released

A cut is not local

Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.

material the bar is the draws whose letters do not contradict themselvesa loop of panels is a proof that no flat folded state exists, and it costs one passone square twist39 of 409 panels · 12 creasesone hexagon twist40 of 4013 panels · 18 creasesa small square tiling24 of 4049 panels · 72 creasesthe square tiling7 of 4049 panels · 84 creasesthe patch a propagation returns first is not a draw and has no reason to be among these

The tiling the unit could not promise

Every twist on this site carries the same caveat: the unit is verified and the plane is not, because deciding a whole pattern is intractable. There is one thing about a whole pattern that costs a single pass over its crease list, and it says no. The square twist tiling was drawn with a lettering that contains a loop of twenty-eight panels, so the patch on this site had no flat folded state at all — and only seven of forty independent redraws avoid one.

tessellation each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsa loop of 8 panels — no order existsthe arrows are the whole of the test — nothing here asks which panels lie over which

The ring is the loop

The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself.

tessellation

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