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a fold stops working when the stack reaches 3 mm 8 layers 16 layers 32 layers 64 layers 128 layers newsprint 65 µm 520 µm 1.0 mm 2.1 mm 4.2 mm 8.3 mm copier paper 100 µm 800 µm 1.6 mm 3.2 mm 6.4 mm 12.8 mm kami 70 µm 560 µm 1.1 mm 2.2 mm 4.5 mm 9.0 mm washi 40 µm 320 µm 640 µm 1.3 mm 2.6 mm 5.1 mm foil-backed tissue 26 µm 208 µm 416 µm 832 µm 1.7 mm 3.3 mm unryu tissue 18 µm 144 µm 288 µm 576 µm 1.2 mm 2.3 mm thickness measured across the sheet; the smallest feature is a folder's working figure rather than a constant of nature
Paper as substrate
1
The paper had to arrive first
2
A sheet has a size as well
3
Which ceiling is binding
4
A sheet is as large as two arms
5
The paper that will not hold a crease
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8 essays · history
what the construction produced 23 twists, 122 interior vertices turned 24.1° from the tiling's edges pleats 0.068 to 0.068 wide 1.58× smaller once the pleats are taken up every vertex passes all four conditions mountain valley raw edge
Resch
1
Where two twists share a pleat
2
The propagation that never had to work
3
Closing the loops is not folding
4
One number where the corners wanted four
5
Every twist writes an equilibrium
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8 essays · tessellation
20° a crease 0.50 turns of paper nothing touching anything 34° a crease 0.85 turns of paper nothing touching anything 36° a crease 0.90 turns of paper 1 pair through one another 50° a crease 1.25 turns of paper 5 pairs through one another one strip of 10 panels, seen end-on it laps itself at 36.0° a crease, which is where its cross-section closes every panel is the same length in every frame; the only thing changed is how far each crease is turned
Self-contact
1
Paper through paper
2
Closing is not building
3
A collision is an order
4
Two refusals that refuse differently
5
Refused at one lettering
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8 essays · rigid
-100 100 0 0.2 0.4 0.6 0.8 1 how far the vertex is driven stored energy branch one MVMM branch two MVVV both run downhill from the flat state, and end at zero so the energy does not prefer either branch — the noise decides
Self-folding
1
Paper that folds itself
2
One crease decides the sheet
3
Which crease to push
4
The hardest instant
5
Only four creases decide a Miura
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8 essays · rigid
axiom 1 through two points linear axiom 2 point onto point linear axiom 3 line onto line linear axiom 4 through a point, square to a line linear axiom 5 point onto a line, through a point quadratic axiom 6 two points onto two lines cubic axiom 7 point onto a line, square to a line linear the degree each axiom can solve — one of them is why paper beats the compass
The axioms
1
One fold at a time, and there are exactly seven of them
2
Folding beats the compass, by exactly one degree
3
Why the list stops at seven
4
Where the cubic comes from
5
What each axiom is worth
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8 essays · construction
the tree leg 0.62 leg 0.62 body 0.34 arm 0.50 head 0.78 the base the axis 0.62 0.62 0.34 0.50 0.78 the flaps are the tree's edges, at the tree's lengths, all square to one line so the base's shadow along the axis is the tree, and nothing else can be designed this way which is the restriction the circle argument quietly depends on
Uniaxial bases
1
Every flap on one axis
2
A base needs an edge to point at
3
Every pair, not every circle
4
What the condition does not decide
5
The price of a limb is not its length
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8 essays · design
Paper is made in China Paper reaches Japan Paper is made in Europe Folded paper is used ceremonially in Japan 400 yr Paper is folded for amusement in Japan 980 yr The thousand cranes 897 yr The pajarita is folded in Spain 293 yr Paper folding is taught as geometry One fold solves a cubic The diamond pattern in a crushed cylinder The conditions at a flat-foldable vertex The dashed-and-dotted diagram notation The Miura fold A five-pointed star from one straight cut Any straight-line drawing, from one straight cut year of the source 500 1000 1500 2000 the date generally given the oldest source that says so median overrun 201.5 years
Attribution
1
Nothing here is as old as it sounds
2
The name is not the date
3
Fifty years in the wrong language
4
A record is not a proof
5
The patterns nobody owns
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7 essays · history
found rather than designed crushed drink cans, tree bark, deployable booms at every interior vertex sectors 60°, 60°, 60°, 60°, 60°, 60° two courses and four diagonals two of one letter, four of the other why the height is not free a steeper diagonal makes the top sector strictly smallest, flanked by two of the same letter 20 interior vertices · 23 mountain and 56 valley creases mountain valley raw edge
Found patterns
1
Patterns nobody designed
2
The cylinder the pattern chooses
3
The period nobody measured
4
A twisted tube has two heights
5
Four clicks make sixteen lengths
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7 essays · tessellation
Folded paper is used ceremonially in Japan 400 yr Paper is folded for amusement in Japan 980 yr The thousand cranes 897 yr The pajarita is folded in Spain 293 yr Paper folding is taught as geometry year of the source 500 1000 1500 2000 the date generally given the oldest source that says so median overrun 400 years
Two traditions
1
Two traditions and a merge
2
When two of them are first attested together
3
One lost source and the story changes
4
The interval is wider than the number
5
A question the record is too small to answer
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7 essays · history
two kinds of vertex, both forced 16 of degree 4 90°, 90°, 90°, 90° 9 of degree 6 90°, 45°, 45°, 90°, 45°, 45° 40 mountain and 36 valley creases 14.3 sheet-widths of folding mountain valley raw edge
Waterbomb
1
The base that tiles
2
A unit that folds is not a tessellation
3
Thirty-two rules, one object
4
Thirty-two rules, thirty-two pieces
5
The rule that breaks the count
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7 essays · tessellation
0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 flap width, as a fraction of the sheet area showing the two are equal at a third front showing reverse showing total face measured on the folded state at 4 flap widths, and the marks are those measurements
Colour change
1
Bringing the other side to the front
2
Decided before the design
3
Which side arrives
4
One sheet down
5
A corner turned shows the sheet beneath
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6 essays · design
3 equal parts estimated by eye the left-hand divisions are exact — a consequence of the fold, not of care the right-hand ones are a guess, and the error compounds valley mountain
Exact division
1
Dividing without measuring
2
Folding a strip into thirds
3
One crossing, and then another
4
Exact is not accurate
5
The grid a division makes
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6 essays · construction
n φ(n) its prime factors compass one fold two at once 3 2 2 4 2 2 5 4 2 · 2 6 2 2 7 6 2 · 3 8 4 2 · 2 9 6 2 · 3 10 4 2 · 2 11 10 2 · 5 12 4 2 · 2 13 12 2 · 2 · 3 14 6 2 · 3 15 8 2 · 2 · 2 16 8 2 · 2 · 2 17 16 2 · 2 · 2 · 2 18 6 2 · 3 19 18 2 · 3 · 3 20 8 2 · 2 · 2 21 12 2 · 2 · 3 22 10 2 · 5 23 22 2 · 11 24 8 2 · 2 · 2 the 11-gon is the first a single fold misses, and two simultaneous folds reach it the 23-gon is the first that needs more than two, because 22 has an 11 in it
Multifold
1
Two creases at once
2
Seven, and then twenty-two
3
Twenty-two is a floor
4
Each fold needs its own two
5
Counting operations is not counting power
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6 essays · construction
proportion 1.3 two shapes, in turn 1.300 1.538 1.300 1.538 1.300 proportion √2 = 1.4142 one shape, throughout 1.414 1.414 1.414 1.414 1.414 halving turns a proportion of r into one of 2/r, and those are the same number only at √2 the 1.3 sheet is a different shape after every fold; the √2 sheet is the same shape after all of them a square metre at √2 is 840.9 × 1189.2 mm, which is the 841 × 1189 printed on a sheet of A0
Paper proportion
1
The rectangle that keeps its shape
2
One member of a family
3
The proportion a band asks for
4
A construction assumes its sheet
5
A stretch keeps crossings
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6 essays · construction
20 panels, two colours no crease has the same colour on both sides all 12 interior vertices carry an even number of creases the colour is which side of the paper that panel shows when the sheet is folded mountain valley raw edge
Two-colourability
1
The sheet has two sides
2
Even is not enough
3
A contradiction is even
4
The seam carries a sign
5
The band that needs an odd number
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6 essays · flat-folding
the bar is the vertices the drawing has and the list does not The preliminary base 0 9 listed · panels close The Miura fold 0 35 listed · panels close The square twist 0 16 listed · panels close The hexagon twist 0 22 listed · panels close The Yoshimura pattern 0 45 listed · panels close Fold and cut — the triangle 0 11 listed · panels close The tapered corrugation 0 40 listed · panels close The waterbomb tessellation 0 41 listed · panels close the square grid, assembled 0 64 listed · panels close the triangular grid, assembled 12 82 listed · panels 1.73 apart the honeycomb, assembled 18 84 listed · panels 2.00 apart the rhombille tiling, assembled 12 138 listed · panels 1.86 apart the elongated triangular tiling, assembled 5 76 listed · panels 1.73 apart every pattern with a bar has panels that cannot be placed, and every pattern without one places exactly
As drawn
1
The vertex the list does not have
2
Two creases that cross
3
One solution of a search nobody ran
4
The crease the drawing cannot show
5
The drawing does not say what is glued
5 essays · flat-folding
-3 -2.5 -2 -1.5 -1 -8 -6 -4 -2 0 tolerance (log₁₀ radians) fraction inside it (log₁₀) 1 vertex · slope 1.00 2 vertices · slope 2.01 3 vertices · slope 3.01 40,000 random vertices, none of them constructed to fold and none of them folding
Genericity
1
Almost every pattern fails
2
A near miss is nearly as rare
3
Where a sector crosses sixty
4
A region with no lettering
5
How rare a band that folds is
5 essays · flat-folding
0 0.2 0.4 0.6 0.8 0 1 2 3 how far the sheet is closed −ν, so every curve shown is a negative ratio Miura, slant 0.25 Miura, slant 0.42 Miura, slant 0.6 accordion exactly zero the same paper, three behaviours, chosen by the crease pattern alone
Mechanical metamaterials
1
A material made of creases
2
A property you can dial
3
Nothing to average over
4
The property a patch does not have
5
A metamaterial with no edge
5 essays · tessellation
sectors 80°, 55°, 100°, 125° in every one of them, and 4 assignments fold in every one longest ÷ shortest 1.00 footprint 0.806 longest ÷ shortest 3.09 footprint 0.911 longest ÷ shortest 3.33 footprint 0.623 longest ÷ shortest 4.00 footprint 1.782 every one of them folds; their folded footprints differ by a factor of 2.86
Sector angles
1
The lengths are free
2
Where the lemma says nothing
3
The order decides the count
4
One step per panel is a table size
5
An alternating sum of angles
5 essays · flat-folding
2 3 4 5 6 7 8 9 10 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 discs shortfall of the symmetric search 14.6% 1.7% 0.0% -0.0% 5.6% 0.5% 10.8% 10.4% 4.7% symmetry: mirror · both searches at 90 restarts neither number is a proved optimum — this compares two searches
Symmetry
1
When symmetry costs
2
The shapes the optimum has
3
The symmetry the letters cannot keep
4
The symmetry a gluing adds
5
Equal effort was counted in steps
5 essays · design
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