Origami & Folding
what one sheet can do
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The figure library — page 9

Every picture here is generated from code at build time. This is every generator, ordered by how many essays call it, each rendered at its defaults.

rigid-limits

4 essays — Panels instead of paper · Paper through paper · The only pattern that moves · A mechanism that closes on itself · 6 claims checked

A strip with no vertex in it, rolled until it meets itselfA strip of 10 panels creased along parallel lines, seen end-on at 4 crease angles. It has no interior vertex, so every local condition the subject checks is satisfied with nothing anywhere to evaluate. The panels picked out in the last frames are the ones passing through one another, which no condition on a neighbourhood could ever have reported.20° a crease0.50 turns of papernothing touching anything34° a crease0.85 turns of papernothing touching anything36° a crease0.90 turns of paper1 pair through one another50° a crease1.25 turns of paper5 pairs through one anotherone strip of 10 panels, seen end-onit laps itself at 36.0° a crease, which is where its cross-section closesevery panel is the same length in every frame; the only thing changed is how far each crease is turned

rule-object

4 essays — Thirty-two rules, one object · The count counts labels · Thirty-two rules, thirty-two pieces · The rule that breaks the count · 2 claims checked

Five hundred and twelve rules, one objectThe repeating rules for a waterbomb tessellation, counted at each stage: every rule, the ones that pass the conditions on a small patch, the ones that pass on a patch containing every kind of vertex, and the number of distinct folded objects those produce. The last number is one.counting rules and counting objects are different measurementsrepeating rulesnine binary choices, one per crease of the repeating unit512pass on a small patchevery vertex of a two-by-two patch satisfies every condition56pass on a larger oneand on a patch that contains all four kinds of vertex32folded objectscounted by comparing the folded panels, not the letters1

ruling-envelope

4 essays — Where the rulings run out · The gap between two curves · Only one side can run out · An angle that turns faster than the crease · 17 claims checked

The crease, its rulings, and the line where they crossA curved crease in red; the rulings of the folded surface leaving it at a fixed angle; and in magenta the curve where consecutive rulings meet. Each ruling is drawn only as far as that curve, because past it the surface has folded through itself.the crease, its rulings, and the line where they crossan elliptical crease, rulings at 52° from the tangentthe shortest ruling0.0998the tightest radius0.1274their ratio0.7833nothing on the crease pattern marks this line, and no amount of paper moves it

symmetric-packing

4 essays — When symmetry costs · The corner is worth four times the middle · The shapes the optimum has · Equal effort was counted in steps · 11 claims checked

What the symmetry is worth, count by countHow far the symmetric search falls short of the free one, as a fraction of the radius, at equal effort. Above the line the constraint costs something real; below it the constraint has made the search easier than the freedom did.2345678910-0.15-0.1-0.0500.050.10.15discsshortfall of the symmetric search14.6%1.7%0.0%-0.0%5.6%0.5%10.8%10.4%4.7%symmetry: mirror · both searches at 90 restartsneither number is a proved optimum — this compares two searches

tree-on-a-grid

4 essays — Spelling a tree on a grid · The other grid · Ninety-nine in a hundred pass · Rounding in the cheap direction · 2 claims checked

Rounding a design is not rounding its limbsThe worst limb error against how fine the grid is, for the obvious rounding and for the best whole-number version there is. The obvious one is not monotone — a finer grid can round worse — and the best one is flat over a wide range, because the same coarse set of whole numbers goes on being the best answer.42:3:4:363:4:6:583:4:6:5125:7:11:8165:7:11:8248:11:17:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution

crimp-gap

3 essays — Crimp it away and ask again · The whole alphabet of a grid · Which vertices are the random ones · 8 claims checked

Where the conditions stop being the answerFor each row, how many mountain-and-valley labellings satisfy all four local conditions, how many of those have a flat folded state, and the difference. At degree four the two columns are the same number; above it they are not.pass all fouractually foldthe gapdegree 4373 vertices14921492nonedegree 6329 vertices32082632576degree 858 vertices1440928512the gap is what a checker built from the four conditions would certify and a folder could not make

crimp-ladder

3 essays — Crimp it away and ask again · A short reason to say no · The cost is in the coincidences · 2 claims checked

One vertex, folded away two creases at a timeA vertex of degree six, and the sequence of smaller vertices the crimp reduction takes it through. Each step folds the sector strictly smaller than both its neighbours away between them, which removes two creases and merges three sectors into one. The shaded wedge is the sector about to go.the vertex on the paper6 creases0 crimps, and what is left is one straight crease with one letterthe four conditions do not all hold · a stacking does not exist

crimp-tree

3 essays — A tie is not a decision · Which vertices are the random ones · The other grid · 7 claims checked

What a tie costsThe vertex of equal sectors at each degree, with how many smaller vertices the reduction visits before it answers, against how many crimps the answer actually needs. The gold bar is the work and the green mark is the necessity.degreevertices visited per letteringcrimps needed48 of 16 fold32630 of 64 fold1038112 of 256 fold41410420 of 1024 fold2065121584 of 4096 fold12376The work grows by a factor of about 6.0 for every two creases added; the necessity grows by one.

crumple-growth

3 essays — Every facet is a layer · The decision a crumple has taken · The crumple keeps its options · 2 claims checked

Folding adds and never subtractsInterior vertices, facets and total crease length against the number of random folds, averaged over five seeds. Each curve rises and none of them turns over: a fold can only add creases, so the pattern gets finer at every step and the facets between the creases get smaller. Every sheet in the sweep passes the flat-folding condition at every vertex.24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.6e-1 to 4.0e-3 of the sheet

deficit-two-ways

3 essays — A crease carries no curvature · The test measures the rim · A straight tuck is a cone point · 3 claims checked

The same curvature, kept in one place or spent everywhereRemoving a wedge and closing the gap makes a cone: all of its curvature sits at the apex and the rest of the surface is flat, so a walk that avoids one point measures nothing. A growth field carrying the same total has no special point at all. Folding concentrates; growth spreads.a crease — concentrated at a pointdeficit 0.5236 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.5236 rad, none of it anywherecurved at every pointa 30° wedge removed, against Ω = 1 − 0.0400 r² · same total curvature, 0.5236

edge-flap

3 essays — The corner is worth four times the middle · A hole is cheap paper · Spending the cheap paper · 6 claims checked

A flap in a corner costs a quarterThree flaps of the same length on one sheet: one in the middle, one on an edge, one in a corner. Each consumes every point of paper within its own length of it, but only the paper that is actually there — so the same flap costs a whole disc, half of one, or a quarter, and the boundary is the cheapest place to stand.1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it

four-questions

3 essays — Four questions about one sheet · What a checker cannot check · A no costs more than a yes · 1 claim checked

Four questions, one stripThe same crease pattern asked four different things. Deciding stops at the first answer, counting cannot stop at all, listing pays for the answer as well as the search, and asking whether a machine can make it is a question about sequences rather than about states — a different search over a different space.6 creases, 7 segments, assignment MVMVMVDoes it fold flat?at most 5,040 orderings, and it may stop earlyyesas far as the first legal oneHow many ways?every one of them, because the last is as likely as the first15,040 orderingsWhat are they?the same search, paying a second time for what it keeps1 stackings, written out5,040 orderings, and the answer as wellCan a machine make it?a different search, over sequences of folds rather than over stackingsyes1,350 statesthe four are not four difficulties of one problem — they are four problemsthe cost is work rather than time — a clock reading would differ on every build