Origami & Folding
what one sheet can do
Concept

Anisotropy — where it appears

Having different properties in different directions. Paper has it because its fibres lie mostly one way, and no theorem in this subject mentions a direction, so nothing here can advise on which way to lay a pattern.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

24682468times shorter across the sheettimes shorter down itaccordionMiuraleaf corrugationYoshimurasquare twistwhat each one doesaccordion8.00× across, 1.00× downMiura2.73× across, 1.75× downleaf corrugation3.57× across, 1.22× downYoshimura4.00× across, 4.00× downsquare twist1.52× across, 1.52× downthe dashed line is a directionthe fold never touchesone direction untouched: 1 of 5 · equal both ways: 2 · neither: 2the single number a corrugation is usually quoted by is these two multiplied together

A shrink is two numbers

How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.

tessellation · Shrinkage the ratio of each column's cosine to the first column'scol 1col 2col 3col 4col 5the sheet that folds1.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3one row moved by 14 per cent1.0000-1.08221.0000-1.08221.00001.0000-1.13971.0000-1.13971.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3worst disagreement between rows: 4e-16 before, 0.0575 after

Where an error goes

A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.

rigid · Tolerance patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions54% / 46%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 11%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45°

The fifth thing that is not true

Four idealisations underlie every theorem here and each has had an essay. There is a fifth and it has never been named, because it is invisible in exactly the way the others are not: paper has a grain, no theorem in the subject mentions a direction, and so nothing in the whole apparatus can tell a folder which way up to lay the pattern down.

material · Idealisation the Yoshimura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters36283224panels29202416V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

Which pair is glued

A cell's two cylinders have the same Euler number, the same amount of rim and the same name. On a symmetric drawing they have identical counts of letters, panels and vertices — and searching them costs twenty-four nodes one way and eighty-five the other. Half the rim is a description of the topology and not of the object.

complexity · Search order 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

Named alongside it

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

TwistBoundaryBounding boxClosureCorrugationCrease assignmentCrease patternDeploymentDesignEquilibrium stressError propagationFold angle

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