Pleat — where it appears
Named by 15 essays across 2 fields — each of them below, with the objects they name alongside it.
A square that turns
A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.
Any tiling makes a twist
A twist tessellation is usually drawn, admired and copied. It can be derived instead: hand the construction any tiling of the plane and it returns a crease pattern that folds flat, with the twist polygons' shapes forced by the tiling's own angles and nothing left to choose but how large and how turned.
Where two twists share a pleat
Every twist tessellation the tradition draws has one size of twist, because every tiling it is drawn on has one kind of vertex. Hand the construction a tiling with two, and the pleat between a large twist and a small one turns out to fix their sizes exactly — three to one, and nothing else folds.
The propagation that never had to work
The twist construction carries one equation per edge of its tiling and propagates the twist sizes outward from a seed. On every tiling anybody has drawn a twist on, every one of those equations is satisfied trivially — both ends of an edge read the same two numbers, because a regular polygon has one interior angle. The construction has been running and doing nothing, and the one tiling where it did something is the one whose tiles are not regular.
The gap between two curves
The rulings leaving a curved crease are not parallel, so they cross, and the surface exists only as far as the first crossing. That bound is usually read as a limit on how far a design extends outward. It is not: the paper between two curved creases has to be reachable from both, so the bound bites hardest where the circles are smallest, and a concentric pleat has a hole in the middle that no sheet size removes.
The ring is the loop
The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself.
A tuck keeps what a gore cuts
A flat disc gathered into a spherical cap has more circumference than the cap, and a gore removes the excess while wet-folding stretches it away. A tuck folds it under, which keeps the sheet whole and turns the excess into thickness. At the rim of a gathered cap the paper is α ⁄ sin α sheets thick on average — π⁄2 for a hemisphere — and a simple tuck is three, so single tucks reach a cap of 130.6° before they run into one another. And because a sphere's circles fall short of a plane's as the cube of the radius, a tuck that follows the sphere widens as the cube too: its edges are curves.
A straight tuck is a cone point
A tuck with straight edges hides length in proportion to how far past its start it has gone, which is a cone's law and not a sphere's. Started at the centre, straight tucks make a cone. Started at several radii, they hide length in a broken line that follows a sphere's cubic, and the worst shortfall falls as the square of the number of starting radii: 36.9 per cent of the rim's hiding from one start, 12.3 from two, 3.3 from four, 0.8 from eight. Every start is three creases at a point, which is a vertex that cannot fold flat — and it is exactly where the gathered sheet's curvature goes.
Crowd the tucks toward the rim
Straight tucks started at several radii follow a sphere's hidden length in a broken line, and evenly spaced starts leave a worst shortfall that falls as the square of their number. Evenly spaced is not the best spacing. A sphere's hiding bends hardest near the rim, so the best starts crowd outward — on a hemisphere, four of them at 0.36, 0.60 and 0.80 of the radius — and leave 38 per cent less error than four evenly spaced. As the count grows the saving closes on 42 per cent, a limit set by the square root of how the sphere's hiding bends; on a shallow dish it approaches five ninths. To follow a hemisphere within one per cent takes six rings of tucks instead of eight, and within a tenth of a per cent eighteen instead of twenty-four.
Crowding outward costs almost nothing
Placing the tuck starts for equal error crowds them toward the rim, where the gathered paper is already at its thickest, and the obvious worry is that the accuracy is bought with depth. Measured, it is not: on a hemisphere the crowded placement's deepest start sits in 3.37 sheets against the even placement's 3.32, because a start is three sheets of its own and the gathering beneath it is only one and a half.
Three answers, one count
Seams, curved creases and a few per cent of stretch are the three ways round the sphere, and a tuck is a fourth. All four dispose of one quantity — the excess circumference a flat disc has over the sphere's circle — and all four dispose of it by dividing the circle. So the number of divisions needed is the same whichever answer is chosen: nineteen for a hemisphere in a material that gives two per cent, eight at five, four at ten. What differs is what a division costs, and one of the four runs out.
Where a ring of divisions belongs
A pattern that divides the circle everywhere as finely as its rim requires is over-divided for most of its radius, because the excess grows from nothing. Putting a ring of new divisions in wherever the residual would otherwise pass what the material takes gives seven rings on a hemisphere at five per cent of stretch, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out — and the first of those sits where a completely different criterion put its first tuck start.
Closing the loops is not folding
The twist construction propagates one equation along every edge of a tiling, and it can only work where the equations agree round every loop. Asked which irregular tilings pass, a linear map gives a clean answer: the square grid, the triangular grid and the honeycomb pass under every shear and stretch tried, because each edge has a half-turn symmetry that makes its equation exactly one at both ends, and a half-turn survives any linear map. The rhombille passes only as drawn. But passing is not folding. On every one of those images — including the ones whose loops close exactly — the construction produces a pattern that fails the angle condition at every turn tried. The loops were a necessary condition all along, and the construction needs something the tilings' images do not give it.
One number where the corners wanted four
The twist construction gives a vertex a single side distance, and every account of these patterns does the same — it is what rotate-and-shrink means. The conditions never asked for it. Written out, the corner condition is one linear equation per pleat crease in the distances taken one per edge, so a degree-four vertex carries four unknowns against two independent equations. Given them back, the twelve sheared and stretched tilings that refused to fold all fold.
Every twist writes an equilibrium
Divide each side of a twist polygon by the length of the edge it faces. The polygon closing says those numbers, weighted onto the edges, balance at the vertex; the pleat matching says the two ends of an edge agree on the number. Together they are a positive equilibrium stress — the thing a tiling has when it is the plan of a spider web — and the construction has been writing one at every vertex without being asked for it.
Named alongside it
The objects these essays reach for when they reach for this one.
Developable surfaceTwistGaussian curvatureTilingKawasaki's theoremTessellationGoreLayer countAngle deficitConeConstructionIsometry