Anisotropy — where it appears
Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
The folded strip lies at its own slant
The two directional factors read a folded state against the axes the pattern was drawn on, with its first panel held where it was drawn. A flat-folded Miura is not lying along those axes: it is a strip of fixed width laid at exactly its own slant to them, and the leaf corrugation's strip lies at exactly its zigzag. So the Miura's cross factor changes with the number of columns — 1.748 on four, 0.948 on ten, at one slant — and on the leaf, on every Miura with an odd number of columns and on the twists of three and six sides, it changes with which panel is held. Read without axes, both families compact least at 0.80 radians, not at the 1.05 the box on the axes reports.
A dial can turn the fold over
A twist tessellation drawn on a sheared square grid folds to a sheet that draws in 4.4 times as much one way as the other, and the obvious guess is that the number belongs to the tiling and the shear. It does not: the turn and the pleat angle move it from 1.5 to 383. Between pleat angles of −0.348 and −0.161 the collapse's area changes sign, and the twists fold to a mirror image of their own arrangement.
A cut cannot tell which way a leaf grew
Let a sheet grow faster in one direction than the other, and a cut still reads a flux across its edge — but of a field whose parts are how fast the growth along each direction changes across it. Growth that varies only along its own direction curves nothing. And a whole family of growths shares one such field: a disc grown only along its midrib by 1 + a y², one grown equally every way by e^(a y²), and one grown more across than along all hold the same curvature in every piece, though one grows to twice the area of another. No cut, of any shape, can tell which way a leaf grew.
Growth that only turns
Grow a sheet by the same two stretches at every point — 1.4 along one direction, 1 across it — and let only the direction change from place to place. No rate varies anywhere, and the sheet still cannot lie flat. A cut reads two whole turns' worth of geometry: the number of times the direction turns round it, less the circulation of a field built from how fast the direction turns moving along itself and moving across itself. Directions radiating from a point put all the curvature at that point, a cone of 2π(1 − 1⁄1.4); directions running round it make an anticone; a spiral pitched forty degrees from the radius holds nothing at its centre; and a direction that turns steadily across the sheet curves it in stripes of alternating sign.
Named alongside it
The objects these essays reach for when they reach for this one.
ShrinkageTwistFootprintGaussian curvatureMeasurementTessellationTilingBoundaryBounding boxClosureConeCorrugation