Miura — the series
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One vertex, repeated
Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.
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A sheet with one freedom
A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.
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The corrugation that curves
A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.
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The Miura folds two ways
One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.
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The paper a pattern asks for
A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.
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A corrugation agrees with itself
A Miura fold of forty-eight panels and a twist tessellation patch of forty-nine have almost exactly the same number of independent closed chains for their letters to contradict themselves round — thirty-five against thirty-six. Sixty-four per cent of the Miura's drawn letterings are consistent and thirteen per cent of the patch's. A Yoshimura at thirty-three chains manages ninety-three. The room to fail sets the scale; the construction decides where in it a pattern lands.
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Sixty-four rules, sixteen fold
The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.
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The loop is in the rule
Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.
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A corrugation never backtracks
As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.
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The corrugation that closes on itself
A Miura cell crosses one crease per period in one direction and four in the other. So one of its two directions has a parity condition that half the sizes fail, and the other direction cannot fail at any size — the same sheet, the same drawing, and two gluings that behave completely differently.
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One node per panel, with the rim gone
A rectangle of repeating pattern cut out of the plane costs exactly one node of search per panel, on every family and at every size. Take the rim away and the total falls and the cost per panel rises, because the letters that were removed were the ones that could not be wrong.