Origami & Folding
what one sheet can do
Concept

Shrinkage — where it appears

How much smaller a folded sheet is than the paper it came from, as a ratio. It has more than one honest answer: the fraction of paper not taken up in pleats depends on the tiling, and the fraction of plane the folded state covers is set by the pleat and is the same on every tiling here.

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

what the construction produced7 twists, 60 interior verticesturned 17.2° from the tiling's edgespleats 0.123 to 0.123 wide1.76× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge

The dial that decides nothing

Turn a twist tessellation's angle from one fence to the other and every measurable thing about it changes: the smallest sector goes from 88 degrees to under one, the pleats swallow a quarter of the sheet and then almost none of it, the folded footprint changes by a third. The number of ways it can be creased does not change at all — sixteen, at every angle tested — because the lemma reads which sector is smallest and never how small.

tessellation · Twists one similarity, five tilingslong: a cell of the flat sheet · short: where it lands foldedthe square grid ×0.41037344the triangular grid ×0.41037344the honeycomb ×0.41037344the elongated triangular tiling ×0.41037344the rhombille tiling ×0.41037344turned 36.62°, the same on every onethe scale is a property of the pleat, and the tiling does not enter it

Folding it flat is one similarity

Where a cell of paper goes when a twist tessellation collapses is a scale and a turn: multiply the plane's lattice by 0.410373441 and rotate it by 36.62°. That is the answer on the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the rhombille alike, agreeing to eight figures — while the collection's other answer to how much smaller it gets gives those five tilings five different numbers.

tessellation · Shrinkage the folded period of the Yoshimuradrawn periods across the top123456the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyesa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

The turn a column costs

The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.

tessellation · Shrinkage the folded period of the Yoshimuradrawn periods across the top1234567the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyes240°yesnoa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

The period nobody measured

Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.

tessellation · Found patterns how much more the folded sheet draws in one way than the otherthe ratio of the two principal factors of the collapse, fitted to the twists' positions before and after foldingas drawnshearedstretchedgeneralthe square grid1.00004.41391.00003.5417the triangular grid1.00001.00001.00001.0000the honeycomb1.00002.81795.38555.0597the rhombille tiling1.00002.81795.38555.0597the elongated triangular tiling1.00004.41391.00003.5417one is a similarity — the folded sheet is the flat one scaled and turned, with no direction preferred

The sheet draws in crooked

Every twist tessellation measured here has collapsed by a similarity: the folded sheet is the flat one scaled and turned, the same way in every direction. The patterns that exist on sheared and stretched tilings do not. Ten of the fifteen images fold by a map with two different principal factors, up to five and a third to one — and the prediction that said which ten, made from the weights the pattern writes on its edges, is wrong in both directions.

tessellation · Resch each family's dial, run, in the plane of the two directional factorsmeasured on the folded state of every pattern, not predicted from its rule1234561234along the patternacross itthe accordionthe square twistthe Miura foldthe Yoshimuraa point on the lower edge leaves one direction alone; a point on the dashed diagonal draws in equally both ways

The plane the five points were in

Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.

tessellation · Shrinkage the cross factor of a leaf corrugation against its zigzag anglemeasured off the folded state's own extent, at every angle drawnno change0.600.680.760.840.921.00zigzag angle, radians0.981.001.02below one between 0.7 and 0.88 radians, at worst 0.98555 — and the areal factor never falls below 1.360

The direction that gets longer

A shrink factor below one is a direction in which the folded sheet is bigger than the flat one, and the leaf corrugation has one. The cause is not the taper and not the angle: a corrugation's folded extent across its own creases is a constant of the cell, the same number at two rows and at ten, so the cross factor is the sheet's height divided by a fixed length — a straight line through the origin that crosses one at a definite row count.

tessellation · Shrinkage

Named alongside it

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

TessellationFolded statePeriodicityMeasurementSymmetryTilingTwistThe folded periodFootprintGluingUnit cellAnisotropy

All concepts