Measurement — where it appears
Named by 31 essays across 7 fields — each of them below, with the objects they name alongside it.
Which vertices are the random ones
Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.
Four ways to draw a pattern
Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.
Twelve creases a micrometre long
A patch this collection has drawn for a long time carries a hundred and forty-two creases and a hundred and thirty arcs, and nobody had asked what the other twelve were. They are fragments left where the clip caught a pleat almost exactly at a corner — between one and nine micrometres long on a printed sheet, at one turn angle out of four, and it is the turn the collection prints.
Four easy patches and one that is not
Run the same search a hundred and twenty times on each of five tessellation patches, changing nothing but the order the letters are tried in. Four of them answer in between twenty-five and fifty-three steps every single time. The fifth answers in eighty-four steps at best, a hundred and sixty-six in the middle, and does not answer at all in forty-eight runs of the hundred and twenty.
Stopping is cheaper than finishing
A search whose cost varies by a factor of two hundred with nothing but the order of its guesses should not be waited out. Give up after a hundred steps, reseed and start again, and the whole job costs five hundred and twelve steps in expectation; run each attempt to twenty thousand and it costs sixteen thousand two hundred and ninety-one. Patience is thirty-two times more expensive than impatience.
Four populations with nothing to separate
This collection keeps four standing populations of crease patterns to test its machinery against. Twenty-eight patterns, sampled forty times each for a lettering that agrees with itself and then searched for one — and on every single member the two methods return the same verdict in the same breath. The patterns that separate them are in none of the four, and the reason they are not is what the populations are for.
Rare is not hard
Crumple a sheet deeper and the share of its labellings that agree with themselves falls from thirty-four in forty to eleven. The number of steps a search needs to find one of them does not move at all: it stays at about one per panel, with no backtracking, the whole way down. How often an answer turns up at random and how much work it takes to find one are different quantities, and a crumpled sheet is where they come apart.
The third fold cannot be listed
Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.
A population nobody chose
Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon measured for so long belongs to the one tiling the grid leaves out.
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.
The shortest crease is not a crease
A crease pattern's density is usually quoted as total crease length over sheet area, which treats a metre of folding as a metre whether it arrives as one long line or ten thousand short ones. Reading the lengths individually instead finds twelve creases on a printed patch that are shorter than a wavelength of light.
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.
A reference on a sheet with no corner
Every construction in this subject begins from the sheet's own boundary: two edges meet at a corner, a corner is a point, and a point is what an axiom takes as input. A cylinder has two circles of edge and no corners at all, so a construction on one has nothing to start from and the seam is not a mark.
Three marks see nothing
The measurement a rim cannot make is an interior one, and the obvious interior measurement — two marks a known distance apart, measured again after growth — cannot detect curvature at all, because a uniformly enlarged sheet changes that distance and stays flat. Three marks cannot either: any three distances obeying the triangle inequality are the sides of a flat triangle. Four marks give six distances, and six distances are not free on a flat sheet. The growth profile a rim measurement reads as flat misses by three and a half per cent with four marks at the rim.
A cut reads a slope
A rim measurement returns one number for a whole grown disc, and a family of growth patterns share it. Cut the disc in a circle and the piece inside has a rim of its own, and its reading is minus two pi r times the slope of the log of the growth at the cut — so a cut reads a slope, a set of cuts reads the slope at each radius, and the slopes add up to the growth profile itself. The one thing no cut can recover is how much the whole sheet was enlarged, which is the one kind of growth that curves nothing.
A length needs a scale
These essays measure crease length, and a crease length is a length in the pattern's own coordinates. Six of the eight printed patterns are built on a unit square, so their coordinates are sheet widths and the distinction never arises. Two are not — a Miura laid out as six cells of unit width spans 6.37 — and on those two the shelf multiplied by the printed size without dividing by the width. The Miura's folding length was reported as 6,679 millimetres and is 1,049, and the same pattern's printable sheet has carried the right number all along.
The density a paper allows
Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.
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.
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.
The route, not the sheet
Every cost measured for a glued sheet has been one number from one branch order, and a backtracking search's cost belongs to the pair. Asked under eight orders instead of one, a cut cell's cost barely moves — 42 to 55 nodes — while the torus over the same drawing runs from 69 to 24,636, with one order giving up entirely. The glued sheet's best order costs less than twice the cut sheet's, so most of what a single order charged to the gluing belongs to the route through it.
A stub is never alone
A crossing is a crease running past another and it has a depth. A stub is a crease that simply stops, and it has one too — how far from the rim it stopped, which is also how much shorter than a crease it is. Measured across a hundred and twenty drawings: sixty-six stubs, from 0.27 mm to 35 mm at printed size, every one of them paired with another at exactly the same distance, and not one on a drawing that did not already have a crossing.
Two faults, not four
A drawing departs from its crease list in four named ways — a crossing, a stub, a junction and a fragment — and a checker tests for all four. Counted side by side on two hundred and forty drawings, two of the tests find nothing the others do not. No crease anywhere ends on another crease: every junction the reading finds is a crease meeting the rim, which is where creases are meant to end. Every stub is on a drawing that already has a crossing. What is left is two independent faults, and the clipped construction, which makes no crossings, still makes the second — creases a thousandth of a millimetre long, in pairs.
The cheapest route crosses later
A search for a consistent lettering has a threshold: below it the letters propagate and the cost is a third of a node per crease, above it the search backtracks and the cost explodes. The threshold was measured with one branch order. Measured with eight, the cheapest route never starts searching before the typical one, and on most sheets it starts a period or two later — so part of every threshold on the record belongs to the route. And the one cut sheet past its threshold, the rhombille's, spreads across nearly three orders of magnitude of cost, which moves the spread off the gluing and onto the threshold.
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.
Every panel holds a frame
A flat-folded pattern can be put down on any of its panels, and on some patterns the pair of shrink factors changes with the choice. Holding a panel does not move the paper so much as turn the frame it is measured in, by twice the alternating sum of the crease directions crossed to reach that panel — a number Kawasaki's condition makes independent of the route. So the frames are read off the crease pattern before anything is folded. A pattern with one frame gives one answer whatever its folded shape; the five-sided twist has five frames and five answers; and the four-column Miura agrees on every panel although its folded strip is not its own mirror image, because a box cannot tell a shape from its half-turn.
A circle sees only the average
A cut through a grown disc was found to read a slope: the total curvature inside a circle is minus two pi r times the slope of the log of the growth there, and a set of circles gives the growth back up to one uniform enlargement. That rested on growth arranged in rings. Drop the rings and the integration by parts survives unchanged — a cut of any shape reads the flux of the log of the growth across it — but what a circle reads changes completely. A centred circle sees only the average of the growth's Laplacian round each ring, so a leaf grown at its margins reads, circle for circle, exactly like a disc grown at its rim. And every cut of every shape is blind to any part of the growth whose log is harmonic, which is precisely the growth that curves nothing: a sheet grown e to the ax, faster toward one end, changes area by a factor of three and a half across its width and lies flat.
The route built from the gluing
A search for a consistent lettering of a glued sheet costs a third of a node a crease on its cheap routes and thousands on its dear ones, and an account was offered of which routes are cheap: the ones that take first the creases whose choice propagates through the gluing. That account can be turned into a route rather than tested on samples, and there are two ways to read it. Built the first way, a route finishes nearly twice as many searches past the threshold as a drawn one does, and on the rhombille torus — where none of eight drawn routes finishes inside twenty thousand nodes — six of eight built routes do, the cheapest in 1,183. So the rhombille's difficulty belongs to the route. But which reading works changes with the tiling, and on the elongated tiling, where every drawn route finishes, the built ones lose searches.
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.
No placement cleans every drawing
Five of a hundred and twenty clipped twist patches carry fragments — creases a fraction of a millimetre long, left where the sheet's edge cut through the pattern — and the proposed repair was to nudge the sheet. A fragment turns out to be a corner of a twist polygon lying within a seventh of a millimetre of the rim, and moving the pattern 0.127 mm clears any one drawing. No single movement clears them all. Laid at random, a sheet puts a fragment on one drawing in nine; of twenty-four random placements the cleanest still leaves three drawings with one, and the centred placement every drawing was made at leaves five. A fragment is not a fault of the construction. It is the threshold meeting a placement, and the only clean set of drawings is one placed drawing by drawing.
The torus keeps its tail
Reading why a branch failed removes the heavy tail from the search for a lettering of a cut sheet: on the rhombille drawing at two periods five of eight searches finish becomes eight of eight, and glued along one pair of edges none of eight becomes eight of eight. Glued along both, it stays at one of eight, and it stays there when every refused loop is also remembered. Two accounts of why are refuted on the way: every refusal on every sheet is a loop, so tilings do not differ by how they fail, and a torus's refusals blame no more of the stack than a cut sheet's. What separates them is the length of the proof. Undo twenty decisions from a stopped run and the proof that the letters left are already lost takes 33 nodes on a cut sheet and about 2,400 on a torus, so a jump on a torus has nothing short to read.
A fragment counts creases, not length
A fragment was a piece of crease under a thousandth of the sheet, a threshold chosen for being smaller than a pencil line. Set it instead at the width a crease occupies in real paper, which is the first threshold a folder would recognise, and the rate at which a sheet carries one can be derived from the drawing without placing it: the crease width, times about 2.5, times the number of creases on the sheet. Measured over 1,024 placements on each of 120 drawings it comes out at 1.00 against 0.99 at the width of a crease in kami, with no drawing off by more than a quarter. It reads how many creases there are, and hardly how long they are. It also says what it cannot, which is the clean share: 63% of placements at the width of copier paper, against the 30% independent pieces would leave, because a rim that catches one corner catches its whole row.
Named alongside it
The objects these essays reach for when they reach for this one.
PatchSearch costCrease patternSearchIdealisationBoundaryGluingTessellationFolded stateSamplingShrinkageFootprint