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compass folding 3 2 4 2 5 4 6 2 7 6 the first gap 8 4 9 6 10 4 11 10 12 4 13 12 14 6 15 8 16 8 17 16 18 6 19 18 20 8 21 12 22 10 23 22 24 8 25 20 26 12 n across the top, φ(n) underneath compass: φ(n) a power of two — Gauss, and Wantzel's proof that nothing else works folding: φ(n) with no factor above three, because one fold solves a cubic
Constructible polygons
1
The heptagon a compass cannot reach
2
The eleven-sided one nobody can fold
3
How many polygons a fold reaches
4
What buys the reach costs the accuracy
4 essays · construction
8 letterings fold · 1 piece under any two creases flip two creases anywhere round the vertex, which is the smallest change Maekawa allows MMVVVV MMMMVV MVVVMV MVMMMV VMVVVM VMMMVM VVVVMM VVMMMM sectors 43° · 110° · 121° · 57° · 16° · 13° one piece: every folding is reachable every crease at once: stays inside its own piece
Local moves
1
Walking between two foldings
2
The creases that cannot move
3
The pieces without the list
4
Every move leaves the verdict
4 essays · flat-folding
4 patterns, one folded profile 1/12 2/12 4/12 7/12 2 5 3 1 1/12 3/12 6/12 8/12 2 5 3 1 2/12 4/12 5/12 7/12 2 5 3 1 2/12 5/12 7/12 8/12 2 5 3 1 folded the layer counts under each band are the same in every row, and so are the widths
Inverse problem
1
The shadow does not name the pattern
2
The order does not name it either
3
Which side is showing
3 essays · flat-folding
interior vertices, by number of creases meeting there none 3 odd 59 4 even none 5 odd 32 6 even none 7 odd 1 8 even 8 patterns, 92 interior vertices, and not one of them with an odd number of creases
Vertex degree
1
Nothing meets at three
2
Six creases and the same straight line
3
A knife edge nine decimals wide
3 essays · flat-folding
the waterbomb tessellation's odd vertex degree six, and this site prints nine of them on one sheet every assignment 64 passes all four conditions 30 has a flat folded state 18 12 labellings satisfy every condition the subject has and have no flat folded state
Crimping
1
Crimp it away and ask again
2
A tie is not a decision
2 essays · flat-folding
the pattern the panels, folded sheet 12.000 footprint 1.966 · 6.11 layers on average · 12 at the deepest 1.966 × 6.11 = 12.007, which is the sheet
Conservation
1
The paper is all still there
1 essay · flat-folding
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