Origami & Folding
what one sheet can do
Series

Boundary — the series

13 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 1 row0 interior verticespasses every condition2 rows4 interior verticesfails Kawasaki4 rows12 interior verticesfails Kawasakia pattern that folds is not a pattern whose enlargement folds — the conditions arrive with the interior

    Where the paper stops

    Every flat-folding theorem is a statement about a full turn of paper, so a vertex at the edge of the sheet is subject to none of them. Cutting a patch out of a pattern removes conditions rather than preserving them, and a small enough patch has almost none left.

    part 1 · flat-folding
  2. patternraw edge against crease, by lengthpreliminary base8 panels29.3% rawMiura, 5 by 420 panels20.8% rawsquare twist9 panels26.7% rawYoshimura, 6 by 565 panels6.7% rawthe shaded part is the sheet's own edge; the rest of the outline is creasemeasured with a step of 0.001 of the sheet, and checked across a tenfold sweep of itevery length here is summed over the layers, so a buried edge counts for nothing

    The outline is mostly crease

    The edge of a folded model is what a reader looks at, and almost none of it is the edge of the paper. Measured across five patterns, the sheet's own boundary accounts for between nothing and a third of the exposed edge; the rest is fold, and on a waterbomb tessellation the raw edge does not reach the outside at all.

    part 2 · flat-folding
  3. the edge of the paperMVM40°60°20°60°the same sectors, in a lineMVM40°60°20°60°this lettering folds4 of 8 letterings foldVMV MMV VVM MVMno vertex theorem applies here at all— the sectors do not close, and there is no cycle to alternate round

    The vertices nobody checks

    Every figure on this site is gated on four conditions evaluated at every interior vertex, and the word interior has been carrying the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, not one of them has ever been examined, and the condition that decides them has been available since the second phase of the collection.

    part 3 · flat-folding
  4. the pale bar is the share that folds in a ring, the dark one in a line40 vertices from each population at each degreeangles at random, degree 425% in a ring · 70% in a lineangles at random, degree 613% in a ring · 46% in a linemultiples of 45°, degree 432% in a ring · 83% in a linemultiples of 45°, degree 630% in a ring · 87% in a linemultiples of 30°, degree 429% in a ring · 79% in a linemultiples of 30°, degree 623% in a ring · 70% in a line

    A ring and a line

    A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.

    part 4 · flat-folding
  5. the upper bar is the rim, the lower is the middlethe value is how many other panels an average panel of that kind lies overThe Miura fold18.0 · 21.016 at the rim, 8 away from itThe square twist8.0 · 8.08 at the rim, 1 away from itThe hexagon twist10.0 · 12.012 at the rim, 1 away from itThe Yoshimura pattern61.6 · 64.021 at the rim, 44 away from itThe tapered corrugation19.0 · 22.218 at the rim, 10 away from itThe waterbomb tessellation31.0 · 37.716 at the rim, 36 away from itthe difference is small and it has the same sign every time

    The rim lies over less

    A folded sheet's boundary is usually discussed as the place the theorems stop applying. It is also visible in the pile: a panel carrying a raw edge of the paper lies over fewer of the other panels than one that does not, on every printed pattern that has both kinds — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb tessellation, and never once the other way round.

    part 5 · flat-folding
  6. 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.

    part 6 · flat-folding
  7. the bar is the shortest crease in the pattern, on a scale of powers of tenthe hexagonal patch at four turns of its polygons, everything else heldturn 0.21.4e-2130 creases · every one carries an arc · 2.25 mm on a 160 mm sheetturn 0.357.9e-6142 creases · 12 of them carry no arc · 1.3 µm on a 160 mm sheetturn 0.57.5e-3154 creases · every one carries an arc · 1.20 mm on a 160 mm sheetturn 0.72.8e-2154 creases · every one carries an arc · 4.53 mm on a 160 mm sheetone turn of one patch drops four orders of magnitude below the others, and it is the turn this collection prints

    Twelve creases a micrometre long

    A patch this collection has drawn for a long time carries a hundred and forty-two creases and a hundred and thirty arcs, and nobody had asked what the other twelve were. They are fragments left where the clip caught a pleat almost exactly at a corner — between one and nine micrometres long on a printed sheet, at one turn angle out of four, and it is the turn the collection prints.

    part 7 · flat-folding
  8. one row per pitch, with the printed setting markeda fragment is a crease shorter than a thousandth of the sheetpitchcreasesinterior verticesfragments0.32015466—0.33015466—0.33514260—0.34014260120.34513054—0.35013054—0.36013054—130 creases and 54 vertices is what deleting the fragments was expected to give, and one step of pitch gives it with a sheet that folds

    A patch on a knife edge

    The tessellation patch this collection prints has twelve creases nobody can see. Move the pitch of its tiling by five thousandths and they are gone — and so is a whole ring of twists. The patch sits exactly on the moment a ring of the pattern passes through the edge of the sheet, and the blemish is what that moment looks like.

    part 8 · flat-folding
  9. one period of the square grid's twist tessellationa ring is where a crease leaves and returns on the far side40 crease pieces → 32 creases25 drawn panels → 16 panels16 vertices, every one interiorV − E + F = 0mountainvalleyraw edge

    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.

    part 9 · flat-folding
  10. 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 wayrhombille ×11248 creases become 60 · 24 vertices either wayrhombille ×224192 creases become 216 · 96 vertices either wayrhombille ×336432 creases become 468 · 216 vertices either waythe bar is how many creases the cut divides; nothing else about the two sheets differs

    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.

    part 10 · flat-folding
  11. the Miura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters22182016panels1510128V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

    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.

    part 11 · flat-folding
  12. letters saved by gluing, and the two halves of itthe grid ×121 across + 1 along = 2 · 4 letters cut, 2 gluedthe grid ×242 across + 2 along = 4 · 12 letters cut, 8 gluedthe Miura ×132 across + 1 along = 3 · 7 letters cut, 4 gluedthe Miura ×264 across + 2 along = 6 · 22 letters cut, 16 gluedthe Yoshimura ×164 across + 2 along = 6 · 12 letters cut, 6 gluedthe Yoshimura ×2128 across + 4 along = 12 · 36 letters cut, 24 gluedone comparison says the rim costs something; four say the price is per edge

    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.

    part 12 · flat-folding
  13. the Yoshimura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters36283224panels29202416V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

    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.

    part 13 · flat-folding

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