Origami & Folding
what one sheet can do
Concept

Expected-value — where it appears

A quantity weighted by the chance of getting it, rather than the quantity alone. It matters wherever a design's payoff is uncertain: the fold count that maximises compaction is not the one that maximises compaction times the probability that every hinge works.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

the curve is stop-and-restart; the rule is a constant letter order1001e+31e+41001e+31e+4563 at a cutoff of 10080 nodes, deterministic, nothing to restartexpected nodes in totalcutoff, in nodes

Restarting what cannot be restarted

Stopping a search early and starting it again with a fresh seed costs five hundred and twelve steps in expectation against sixteen thousand for patience. Every number in that is right. The distribution it is right about was made by the search's own coin, and taking the coin out costs eighty — with nothing left to reseed.

complexity · Hardness of folding 200400600800100002004006008001000120014001600mean years ahead of the evidenceresamplingsquoted: 3575%: 15095%: 591spread 13320,000 resamplings of the fifteen entries · five per cent of them fall below 150 and five per cent above 591

The interval is wider than the number

The field's most-quoted statistic — how far ahead of its evidence a popular date runs — is the mean of eight positive gaps, and it is quoted as three hundred and fifty-seven years. Resampling the fifteen entries puts ninety per cent of its weight between a hundred and fifty and five hundred and ninety-one. The interval is wider than the number, the middle gap's interval is a fifth as wide, and the difference is the same two documents the leave-one-out found.

history · Two traditions how many independent sources survive for each claimthe record states this per entry and nothing has ever read the column as a distribution1 source5paper-japan, recreational-folding, senbazuru, beloch, one-cut-star2 sources5paper-china, pajarita, yoshimura, yoshizawa-notation, fold-and-cut3 sources3paper-europe, vertex-conditions, miura-ori4 sources2ceremonial-wrapping, froebela claim resting on one source is one document away from disappearing; five of the fifteen are in that position

A question the record is too small to answer

Five of the fifteen claims rest on one surviving source and five on two, and those two counts are exactly what an estimate of the claims that left no source at all is made of. Applied, it says two and a half are missing. Its ninety-five per cent interval runs from fifteen to thirty-one, re-reading a single entry's source count moves it by a fifth, and its independence assumption is false in the one way documents actually fail — which is what makes computing it worth more than declining to.

history · Two traditions what one document would dono date is invented here — each row asks what the arithmetic would say if one turned upclaimdatedevidenceits gapif a source 200 years earlier turned upceremonial-wrapping12001600400357 → 332 (-25)recreational-folding7001680980357 → 332 (-25)senbazuru9001797897357 → 332 (-25)pajarita15001793293357 → 332 (-25)paper-china1051050nothing changespaper-europe11501056-94nothing changesbeloch19911936-55nothing changesyoshimura19691951-18nothing changesvertex-conditions19891979-10nothing changesyoshizawa-notation19611954-7nothing changesmiura-ori19951970-25nothing changespaper-japan610720110357 → 392 (+35)one-cut-star1776187397357 → 394 (+37)fold-and-cut1922199876357 → 397 (+40)froebel183718381357 → 408 (+51)the statistic is 357 as the record stands; a positive number in the last column is a discovery that makes the field look worse

Some discoveries would make it worse

The field's headline statistic averages the claims dated ahead of their evidence. So a document found for a claim that was nearly right removes a small number from a mean of large ones and the average overrun goes up: finding a source two centuries earlier for the kindergarten entry would take the figure from 357 to 408. The four claims where a discovery helps can take twenty-five years off each, and no single document at any date can bring the number to 250.

history · Two traditions what a finer pattern costs in confidenceeach hinge working 0.999 of the time, against a target of 99 per cent for the whole deploymenthingesthe system openseach hinge needssuccesses to show itand the article itself899.2%0.9987442,385cannot be tested2497.6%0.9995817,154cannot be tested6094.2%0.99983317,885cannot be tested12088.7%0.99991635,769cannot be tested30074.1%0.99996689,422cannot be testedr consecutive successes put a 95 per cent lower bound of 0.05^(1⁄r) on a hinge, and a flight article deploys once

The crease count is a reliability budget

A deployment that needs every hinge to work is the hinge reliability raised to the crease count, so the fineness that buys compaction spends the probability of getting it. At a thousandth of a chance of a hinge failing, sixty hinges give a 94 per cent deployment and three hundred give 74. The fold count that maximises expected compaction is well below the one that maximises compaction — and demonstrating the result takes tens of thousands of successful tests on an article that itself deploys once.

rigid · Deployables one freedom, or severala stuck hinge or a failed actuator loses its own module and nothing elsemodulesall of it opensshare expectedat least 90%at least 75%173.3%73.3%73.3%73.3%272.6%85.2%72.6%72.6%570.4%93.2%70.4%96.0%1067.0%96.1%94.4%99.4%2060.6%97.5%98.7%100.0%5044.8%98.4%100.0%100.0%10027.1%98.7%100.0%100.0%300 hinges at 0.999 each, split evenly · each module's actuator works 0.99 of the time

Splitting a sheet buys area, not certainty

A folded deployable with one freedom needs every hinge and its one actuator, and three hundred hinges at 0.999 each open all the way 73 per cent of the time. Split the same hinges among ten separately driven modules and a stuck hinge costs only its own module: the share of the area expected to open rises to 96 per cent, and the chance of at least nine tenths of it rises to 94. The chance of all of it falls, to 67 per cent, because every freedom added is an actuator added. So freedoms, actuators and reliability trade in a definite way: one freedom is the best design only for a mission that is worthless without its whole area, and for any mission that can live with less, several freedoms win by a margin that no improvement in the hinges matches.

rigid · Deployables

Named alongside it

The objects these essays reach for when they reach for this one.

AttestationDocumentary recordIdentifiabilityPrimary sourceDeploymentIndependent discoveryReliabilityTrade-offActuationCrease radiusDegrees of freedomDeterminism

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