Origami & Folding
what one sheet can do
Series

Flat-foldability — the series

18 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. VMMM60°90°120°90°Kawasaki60° + 120° = 180°90° + 90° = 180°both 180° — satisfiedMaekawa3 mountains, 1 valleysdifference 2exactly 2 — satisfiedangles sum to 360°which is what a flat sheet requiresmountainvalley

    Two conditions at a point

    Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.

    part 1 · flat-folding
  2. MVMwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started

    Why the difference is two

    Maekawa's theorem says mountains and valleys differ by exactly two at every flat-foldable vertex. The constant is not empirical — it is a full turn, and the theorem is about winding rather than about paper.

    part 2 · flat-folding
  3. MVMM40°foldsopposite across the small sectorMMVM40°does not foldthe same on both sidesboth satisfy Kawasaki and Maekawa — the angles and the counts are identical

    The smallest sector decides

    Two assignments can satisfy both flat-folding theorems and only one of them folds. What separates them is a condition about the smallest angle, and it is the first rule in the subject that is not about counting.

    part 3 · flat-folding
  4. 6 interior vertices, every one satisfying both theoremswhat the local tests seeangles at each vertexassignment at each vertexwhat they cannot seewhether layer 3 passes through layer 7whether a flap has room to existwhether the order is consistent everywhereBern and Hayes, 1996: NP-hardso this pattern is checked, not proved

    Local is not global

    Every vertex can satisfy every condition and the sheet still not fold. Deciding whether a whole crease pattern folds flat is NP-hard, which means no figure will settle it and no algorithm will scale.

    part 4 · flat-folding
  5. taco-tacoallowedforbiddentwo folds at the same place may nest or stand clearthey may not interleavetaco-tortillaallowedforbiddena flat layer may pass outside a foldit may not pass through onea crease pattern can satisfy every vertex condition and still break one of these

    Which layer goes on top

    The mountain-valley assignment says which way each crease turns. It says nothing at all about which sheet ends up above which, and that second question is a different object with its own rules — and all of the difficulty.

    part 5 · flat-folding
  6. MVMV123455 segments, 4 creases12345the stack, solvedassignmentsMVMVvalid stacks1decided byexhaustive searchover the orderingsthe folded positions come from the crease spacing; the assignment only decides which way each turn wraps

    A strip is decidable

    Take the same problem down one dimension and it stops being hard. The reason is not that strips are small — it is that overlaps on a line form a chain, and chains cannot contain the cycles that make the two-dimensional question intractable.

    part 6 · flat-folding
  7. the cut line3 straight edgesthe pattern3 skeleton arcs3 perpendiculars1 interior vertexassignments that fold30 of 646 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

    One straight cut

    Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.

    part 7 · flat-folding
  8. degree-4 vertex4 of 1625.0% · 4 creasespreliminary base112 of 25643.8% · 8 creasesmiura 2×28 of 1650.0% · 4 creasesmiura 3×232 of 12825.0% · 7 creasesmiura 3×3256 of 4,0966.3% · 12 creasesevery count enumerated, none estimatedthe share falls as the pattern grows, and the count still rises

    How many assignments fold

    The local conditions throw away most of the ways a pattern could be creased. They throw away a smaller and smaller fraction as the pattern grows, and what survives grows faster than what is discarded — which is why a strong filter is not a decision procedure.

    part 8 · flat-folding
  9. state 0state 1V M M V — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions

    The gadgets that make it hard

    Flat-foldability is NP-hard, and the proof is a construction rather than an obstruction: a machine for turning any satisfiability problem into a sheet of paper that folds exactly when the problem has an answer.

    part 9 · flat-folding
  10. developableKawasakiMaekawabig-little-bigsectors that do not alternatefour creases turning the same waya small sector flanked by one lettera 4×3 Miura, every vertexthe last row passes all four tests at all 6 of its vertices, and passing is not a proofthe tests are conditions at a single vertex; whether the layers can be stacked is a condition on the whole sheetno arrangement of vertex tests decides that, which is what NP-hardness means when it is spelled out

    What a checker cannot check

    Every crease pattern on this site is run past four theorems before it is allowed onto a page, and passing all four proves nothing. The gap is not a bug to be closed: it is the NP-hardness result, arriving as a property of a hundred lines of code.

    part 10 · complexity
  11. 4681012010203040sides of the polygoncreasescreases in the patternperpendicularsskeleton arcsa 12-sided outline needs 36 creases and one skeleton nodea convex outline is the cheap casea reflex corner splits the shrinking front, and this solver refuses those rather than guessing

    What universality costs

    The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.

    part 11 · complexity
  12. what the vertex conditions settle once one crease is chosenpatternsettled, against what is therepreliminary base1 of 81 vertices still choosingmiura 6×41 of 3815 vertices still choosingwaterbomb 4×41 of 7625 vertices still choosingyoshimura 6×51 of 8422 vertices still choosingsquare twist grid3 of 14464 vertices still choosingtriangular twist grid2 of 236104 vertices still choosingKawasaki was settled by the angles before a letter was written; the letters are what is left, and they are nearly all left

    How little the conditions decide

    Local is not global is a statement about sufficiency: every vertex can pass and the sheet still fail. There is a sharper complaint available, and it is about strength. Fix one crease of a tessellation and propagate every condition the subject has to a fixed point: three creases out of a hundred and fifty-eight follow, and sixty-six vertices are still holding more than one answer.

    part 12 · flat-folding
  13. the square twist, sieved three timesevery lettering4,0962 to the 12passes every vertex2566.3% of themletters are consistent2524 force a loop of panelshas a folded state80.20% of themthe bars are on one scale, so the last one is the size of the answer against the size of the question

    The lettering that folds nowhere

    The conditions at a vertex admit 256 letterings of the square twist. Eight of them have a folded state. The other 248 satisfy developability, Kawasaki, Maekawa and the big-little-big lemma at every vertex of the pattern and cannot be folded by anyone — and this site printed one of them for years, at true scale, with instructions to fold it first.

    part 13 · flat-folding
  14. the cut line10 straight edgesthe pattern10 skeleton arcs0 perpendiculars1 interior vertexassignments that fold420 of 102410 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

    One cut for a star

    The fold-and-cut construction here could reach a triangle, a pentagon and a house, and refused everything that turned back on itself, because shrinking an outline with a reflex corner needs an event the shrink did not implement. With split events it reaches a five-pointed star — ten creases through one point, four hundred and twenty letterings that fold, and every edge of the outline landing on one line to a part in 10^16.

    part 14 · flat-folding
  15. 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 loopelongated ×1184 creases80 vertices · every condition holds · no forced loopelongated ×2688 creases320 vertices · every condition holds · no forced loopthe bar is the crease count; the note is what the ordinary checks said

    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.

    part 15 · flat-folding
  16. the composition, and what it has to equal3 reflections, in order[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )=?the gluing map of a Möbius band[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )they agree to rounding, so the band foldsand both turn the paper the same way, so the parity is righton a disc the right-hand side is the identity, which is why nobody writes it down

    Closure is not the identity

    Walk a folded state from panel to panel, composing a reflection at every crease, and come back to where the walk started: the composition has to be the identity. That is the rule everybody states, and it is a special case. On a sheet whose edges are glued the walk does not come back to where it started, and what the composition has to equal is the gluing map.

    part 16 · flat-folding
  17. the composition, and what it has to equal5 reflections, in order[ -1.000 0 ][ 0 1.000 ]+ ( 2.000, 0 )=?the gluing map of a Möbius band[ 1.000 0 ][ 0 -1.000 ]+ ( -2.000, 1.000 )they differ by 4.000 of a width, so it does notand both turn the paper the same way, so the parity is righton a disc the right-hand side is the identity, which is why nobody writes it down

    Parity is not enough

    A Möbius band needs an odd number of creases round it. Give it three, square across the strip, and it does not fold — nor does five, nor seven, nor any odd number at all. The counting argument is necessary and it is not close to sufficient, and the thing it cannot see is which way the creases point.

    part 17 · flat-folding
  18. 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

    The triangle a strip becomes

    A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.

    part 18 · flat-folding

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