Origami & Folding
what one sheet can do
Concept

Independence — where it appears

Whether two pieces of evidence, or two conditions, carry information that does not derive from a shared source. Counting sources as independent when they are not overstates what a record establishes, and treating alignments as independent is what lets their degrees be multiplied.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

distinct fold lines this axiom specifies and no other doesaxiomafter 4 pointsafter 9 pointsafter 565 pointsA1 — through two points08121054A2 — one point onto another08142649A3 — one line onto another4564994A4 — perpendicular through a point001661distinct lines in all1292274300the four operations name 38 folds at the first round and draw 12 lines with them

What each axiom is worth

The list of seven folds is complete, and the proof of that says nothing at all about whether its members are independent or equal. Measured on a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing. Two rounds later the ranking has inverted, and the one that carried the first round is the least productive of the four.

construction · The axioms 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 the year all 5 lineages are on the record, over 20,000 resamplingseach resample draws the lineages with replacement and takes the latest earliest source among those drawnafter 18380.0%no resample can hold a source the record does not183867.1%froebel, 4 witnesses179725.5%senbazuru, one witness17936.4%pajarita, 2 witnesses16800.9%recreational-folding, one witness67.1% return 1838 exactly; the chance a draw of 5 includes the holder is 1 − (1 − 1⁄5)^5 = 67.2%

A resampled maximum can only fall

The joint date of paper folding's five lineages is 1838, and resampling the record twenty thousand times returns 1838 in two draws of three and an earlier year in the rest — never a later one. That is not firmness. It is the chance that a draw of five from five includes one named entry, and it is the reason a resampled interval cannot be put on a maximum at all. Read over all 105 pairs in the record, a joint date inherits the later half of its pair, not the better-attested one: the better in 46, the worse in 35, and a single surviving witness in 28.

history · Two traditions 00.511.520.60.70.80.91failures crowded into one deployment in 10^xchance10 modules, share expected10 modules, at least 90%one freedom10 modules, all of it300 hinges at 0.999 each on average, actuators 0.99 · to the right, the same failures crowd into fewer deployments

Splitting insures only the failures that come alone

Three hundred hinges split among ten separately driven modules were found to open 96 per cent of their area on average against 73 for the same hinges as one mechanism. That gain rests on every hinge failing on its own. Let the same average failure rate arrive in a few bad deployments instead — a cold soak, a crushed stowage — and the one mechanism gains far more than its modules do, because a shared cause makes three hundred hinges behave like one. At one bad deployment in a hundred the lead is gone. A tether that passes a jam to its neighbour erodes it from the other side, and takes the chance of nearly all the area long before it takes the average. Neither touches the chance that everything opens.

rigid · Deployables a 10 by 10 Miura at a slant of 20.1°, with two of its places markedzigzag line 3: 10 hinges, one a rowthey all fold onto one segmentcolumn 3, odd lines: 5 straight hingesthey fold onto one segment toocut between columns into 5 modules

A stack is a shared fate

Hinges that fold onto one segment of a folded stack share whatever happens to that segment. On a Miura every zigzag line puts one hinge from every row in the same place, so a sheet cut between its rows into five modules leaves every place in every module, and one jam anywhere stops all five. Cut between its columns, each place stays inside one module — except at the slants where two over the sine of twice the slant is an even whole number, the very slants that pile the stack evenly, where columns that far apart share places across the cut.

rigid · Deployables a 6 by 5 Miura with zigzag creases slit in pairsa slit ends at two vertices; two slits that meet share one10 of 10 slits20 of 20 interior vertices releasedshare multiplied by 2^20slitreleased, admits every letteringreleased, still refuses somenot released

An array is priced by its vertices

One slit along a Miura's zigzag crease multiplies the share of letterings that pass every local test by four, and one along a straight crease by three. An array of slits is not the product of those factors. Two slits that meet at a vertex release it once, not twice, and on a small Miura the slits' own factors price correctly exactly the 22 of its 128 arrays in which no two slits touch. What prices every array is the product of each vertex's own share — the ring conditions where nothing was cut, the strip rule wedge by wedge where something was. So the cheapest way to release every vertex is a matching: zigzag slits laid in pairs that never meet, half as many slits as vertices. And the count can be made before anything is cut only on a pattern whose vertices decide independently, which the Yoshimura's do not.

material · Kirigami a 6 by 5 Yoshimura and the loops of three that cost ita shaded triangle is a loop of three creases whose correlations multiplyrows 1.732 half-columns tall30 letterings at each vertexevery pair correlated -0.066727 triangles, (−1⁄15)³ eachcounted ⁄ product = 0.99252

What a loop of creases carries

Multiplying each vertex's own share of letterings gives the exact share for a whole Miura, both twists and a tapered corrugation, and three quarters of a per cent too much for the Yoshimura. The difference has an exact form. The count is the product times a sum over loops in the graph of creases, each loop weighted by the correlations its vertices put on the pairs of creases it uses. A Miura vertex correlates no pair at all, so no loop can carry anything. An equilateral Yoshimura vertex correlates every pair by −1⁄15, and its 27 small triangles take the product down by 0.8 per cent before the larger loops pay a little back. Draw the rows taller than √3 and the big-little-big lemma locks three pairs and frees the rest, no loop closes, and the product is exact.

material · Kirigami

Named alongside it

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

Expected-valueAttestationDeploymentDocumentary recordHingeKirigamiLocalityMiuraPrimary sourceActuationAxiomThe big-little-big lemma

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