Origami & Folding
what one sheet can do
Series

Rediscovery — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. what the shell produced14 interior vertices, all alike17 mountain, 40 valley57 creases carrying a letterand it folds flatchecked, not asserted11.0 sheet-widths of crease, chosen by a buckling loadmountainvalleyraw edge

    Found before it was designed

    Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.

    part 1 · history
  2. degree-4 vertex, 60/90/120/90°4 of 164 creases · 25.0% surviveone degree-6 vertex8 of 646 creases · 12.5% survivethe preliminary base112 of 2568 creases · 43.8% surviveand these are only the local tests — a pattern can pass every vertexand still collide once the layers stack, which is the hard part

    The same vertex, found four times

    A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

    part 2 · history
  3. the bar is what the whole job costs if every attempt is stopped thereon the rhombille patch, read off 120 measured runsstop at 10051219% of runs finish by thenstop at 20053033% of runs finish by thenstop at 500105435% of runs finish by thenstop at 1000162442% of runs finish by thenstop at 2000263847% of runs finish by thenstop at 5000569749% of runs finish by thenstop at 100001060450% of runs finish by thenstop at 200001629160% of runs finish by thena run that never finished counts as above every cutoff, so the tail is read conservatively

    The tail was named somewhere else

    The search for a mountain-valley labelling of a tessellation patch costs eighty-four steps at best and does not finish at all two runs in five, and the cure is to stop and start again rather than to wait. None of that was discovered here. The distribution was described in the study of satisfiability solvers in the nineteen-nineties, the restart arithmetic is older still, and what a crease pattern contributes is one more instance.

    part 3 · history
  4. Paper is made in ChinaPaper reaches JapanPaper is made in EuropeFolded paper is used ceremonially in Japan400 yrPaper is folded for amusement in Japan980 yrThe thousand cranes897 yrThe pajarita is folded in Spain293 yrPaper folding is taught as geometryOne fold solves a cubicThe diamond pattern in a crushed cylinderThe conditions at a flat-foldable vertexThe dashed-and-dotted diagram notationThe Miura foldA five-pointed star from one straight cutAny straight-line drawing, from one straight cutyear of the source500100015002000the date generally giventhe oldest source that says somedian overrun 201.5 years

    The cure was named first

    A heavy-tailed search runtime, the arithmetic for cutting it off and restarting, and the reason restarts work at all were established in the study of search between 1993 and 1998. This collection imported all three, and inherited with them the phenomenon they answer — which is that randomising a search's choices is what makes the tail.

    part 4 · history
  5. 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 test imported without its hypothesis

    The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.

    part 5 · history
  6. 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

    Found by people not folding paper

    The shortest strip that makes a Möbius band has a literature, and it is in differential geometry rather than in origami. The two subjects have the same number, they reached it by completely different routes, and neither of them cites the other — which is the fourth time this collection has found that shape.

    part 6 · history
  7. the bar is what the whole job costs in expectation, in nodeson the rhombille patch, over the same 120 measured runs as the fixed cutoffsbest fixed, 100512chosen after seeing the runsunit 132226.30 times the best fixedunit 228545.58 times the best fixedunit 525424.97 times the best fixedunit 1021524.20 times the best fixedunit 2017623.44 times the best fixedunit 508481.66 times the best fixedunit 1005511.08 times the best fixedunit 2006381.25 times the best fixeda unit of one assumes nothing about the runs; every larger unit is a guess at their scale

    What the hindsight was worth

    The best restart cutoff for the one tessellation search with a heavy tail was read off a hundred and twenty measured runs, which nobody running the search could have done in advance. The universal schedule needs no such knowledge, and on the same runs it costs 3,222 nodes in expectation against 512 for the cutoff chosen by looking — a factor of 6.3, which is close to the base-two logarithm of that cutoff, as the theory of the schedule says it should be. A larger unit brings the schedule within a few per cent of the hindsight, and choosing the unit is choosing the scale the schedule was meant not to need.

    part 7 · history
  8. the expected cost of the whole job under rules that learn from their own failures, in nodeson the rhombille patch, over the same 120 measured runs; the dark bar borrows its unit from other patchesbest fixed, 100512chosen after seeing the runsdouble after every failure1820at least 3.56 times the best fixeddouble after every failure, from sixteen1805at least 3.53 times the best fixedgrow by half after each failure1204at least 2.35 times the best fixeduniversal, unit of one32226.30 times the best fixeduniversal, unit from other patches8721.70 times the best fixeda rule that reads only its own failures cannot beat the best fixed cutoff, and cannot know which that is

    A failure teaches a schedule nothing

    The universal restart schedule costs 6.3 times the cutoff chosen by hindsight on the one folding search with a heavy tail, and the obvious repair is a schedule that learns its scale from the attempts it has already made. It cannot. A failed attempt costs exactly its cutoff and reports only that the run needed more, so every rule that chooses the next cutoff from its own failures writes down the same list whatever happens — a fixed schedule in disguise. On the measured runs, doubling after every failure costs at least 3.6 times the hindsight, and growing by half at least 2.4. What does come near is information from outside the run: the universal schedule given the longest search on four other patches as its unit costs 1.7 times the hindsight. The field that supplied the schedule reached the same conclusion, and answered it by watching runs from the inside.

    part 8 · history

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