Found patterns — the series
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Patterns nobody designed
Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.
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The cylinder the pattern chooses
A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.
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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.
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A twisted tube has two heights
Twist a thin cylinder and it falls into a pattern of slanted triangles that clicks between two lengths. The click is usually explained as a fold, but there is no fold. Every crease keeps its own length at exactly two heights, and in between both diagonals have to stretch or shrink. The second height exists only on a course made taller than the one that folds flat, and the Yoshimura's course, the same pattern with no lean, has its second height on the far side of the tube's own axis.
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Four clicks make sixteen lengths
A Kresling tube of four courses has sixteen ways to be at rest, each course shut or open. Pressing and pulling it under one force can reach all sixteen or only five, and the difference is set by how tall the courses were made. Every course shuts at a force that falls as it is made taller, while the force that opens it rises to a peak and then falls. A stack graded below that peak opens shortest first and can only count. Graded above it, it opens tallest first and can be written like a register.
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Clamped, a stack counts in pairs
A Kresling tube of four courses, its ends free to turn, can be pressed and pulled into all sixteen of its words. Clamp the ends so they cannot turn and the courses stop being independent switches, because a course that clicks shut turns by a fixed angle — eighty degrees here, whatever height it was made at — and the others must give it back. With every course of one hand the stack does not click at all, at more than twice the force that shuts any course free. With two courses of each hand it clicks in pairs, one of each hand together, so that their turns cancel: every word it reaches has as many right-handed courses shut as left-handed, and it reaches exactly four. The clamp turns a sixteen-word register into a two-switch one whose every state leaves the ends where they were.
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The Yoshimura folds flat at a few heights
The Yoshimura's row height has always been treated as a free dial below the equilateral: the sectors close, the counts hold, the lemma has nothing to forbid. Asked for an actual stack of layers, its usual lettering refuses nearly everywhere. Folded, its triangles turn round one centre by twice the base angle at each step, and they close on a ring only when that angle is π over a whole number: a triangle at sixty degrees, a square at forty-five, a pentagon at thirty-six. At any other height the rosette spirals, and seven panels are enough to prove it cannot pass itself. Below π over the number of rows plus two, it does not get far enough round to meet itself, so the small patch hides this too.