Origami & Folding
what one sheet can do
Series

Layer multiplicity — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. creases at 0.25, 0.50, 0.75, marked MMM3 legal stackings of 4 segments, read from the bottom of the pile up12341: 2 · 1 · 4 · 3assignment12342: 4 · 2 · 1 · 3taco-taco12343: 2 · 4 · 3 · 1taco-tacothe paper lands in the same place every time — only the order through the pile differs

    More than one way to lie flat

    A crease pattern with its mountains and valleys marked is spoken of as though it named a folded object. It does not. The legal stackings can be counted exactly in one dimension, the count is routinely more than one, and its size is a property of the pattern that nobody quotes.

    part 1 · flat-folding
  2. of the markings that fold, how many folded objects each one makesmarkings that foldexactly one objectthe most any one makesthe preliminary base4 creases · four equal sectors, the first vertex anybody folds881a halved four-crease vertex4 creases · degree four with its two smallest sectors equal — the case the lemma is silent at661the waterbomb tessellation's odd vertex6 creases · degree six, and this site prints nine of them on one sheet18126a Yoshimura vertex6 creases · degree six with every sector equal, and twenty-two of them on the printed pattern30122the preliminary base's centre8 creases · degree eight, and the vertex at the middle of the first base anybody folds112164a vertex at no particular angles6 creases · degree six, drawn from the census and rounded to a tenth of a degree881at four creases the marking names the object; above it, it need not

    One marking, many objects

    A crease pattern with every mountain and valley written on it is spoken of as though it named a folded model. At four creases it does. At six it need not, and at the eight-crease vertex in the middle of the first base anybody folds, a single marking can be folded into four genuinely different objects — same creases, same letters, four answers.

    part 2 · flat-folding
  3. the bar is every legal stacking; the dark part is the ones with no move out of thema swap is legal when the result is still a stacking — the two layers need not be joined by a crease3 segments2 of 6 isolated · 4 legal moves4 segments16 of 16 isolated · 0 legal moves5 segments34 of 50 isolated · 16 legal moves6 segments144 of 144 isolated · 0 legal moves7 segments366 of 462 isolated · 96 legal moves

    Nothing slides past anything

    A marked strip has several legal stackings and this site has counted them at length. Nobody asked whether a folder holding one can reach another by lifting a flap over its neighbour: five hundred and sixty of six hundred and seventy-two stackings have no such move at all, and whether any exists depends on the parity of the segment count.

    part 3 · flat-folding
  4. the bar is the pairs of panels that lie over one anotherThe preliminary base288 panels · 12 rules · an ordering existsThe Miura fold22824 panels · 228 rules · not decidedThe square twist369 panels · 48 rules · an ordering existsThe hexagon twist6613 panels · 96 rules · an ordering existsThe Yoshimura pattern205565 panels · 1187 rules · not decidedFold and cut — the triangle217 panels · 15 rules · an ordering existsThe tapered corrugation28228 panels · 351 rules · not decidedThe waterbomb tessellation92652 panels · 654 rules · not decideda pattern with no bar has no two panels over one another, and its order is not a question

    No height to swap

    A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.

    part 4 · flat-folding
  5. 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

    Consistent is not foldable

    The square twist has 4,096 mountain-valley labellings. Two hundred and fifty-six satisfy every condition at every vertex; two hundred and fifty-two of those have letters that do not contradict themselves; and eight have a folded state. So the cheap proof that reads the letters in one pass accounts for four of the two hundred and forty-eight failures, and the other two hundred and forty-four are refused by a search over orderings that nothing shorter replaces.

    part 5 · flat-folding
  6. the bar is how many DIFFERENT letterings 20 runs returneda coin at every choice1414 of 20 runs found onea constant, with the coin only on the creases no vertex constrains120 of 20 runs found onea constant at every choice120 of 20 runs found oneon the rhombille patch, 157 panels and 282 creases

    One witness or forty

    Taking the randomness out of a search made it three orders of magnitude cheaper in the worst case and cost it thirty-nine of its forty answers. The compromise everybody reaches for — randomise only the choices that cannot matter — recovers four of the forty on two patches and none on the other three, because the diversity was never where it looked.

    part 6 · flat-folding
  7. the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false

    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.

    part 7 · flat-folding
  8. ruling out the square cell's loops, one direction at a timewhat is left splits248 arcs go, 24 remaindirection (1, 0)102 arcs go, 10 remainwhat is left splits010 arcs go, 0 remaindirection (-1, 0)102 arcs go, 10 remainwhat is left splits010 arcs go, 0 remainthe bar is how many arcs are still in play after the step

    The arc that arrived twice

    Which of two panels a crease calls its near one is decided by the order a face walk happened to number them, and the mirrored record is the same relation. Except on one sheet, where it is not — and that sheet turned out to be the one whose folded state comes back the other way up, which is how a duplicate in a graph became a diagnosis.

    part 8 · flat-folding

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