No figure on this site is a drawing that was made once and saved. Each one is a function:
it takes parameters and returns SVG, so the same generator produces the 27° incline and
the 5° incline without either being redrawn.
That is the reason the collection can keep growing without the illustrations drifting apart.
A generator is written once, checked once, and every essay that calls it inherits the same
line weights, the same colour roles, and the same behaviour in dark mode. There are
27 of them so far.
axiom-set
big-little-big
box-pleatingBox pleating Designing on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate. 16 × 16 grid every crease on a grid line, or at 45° which is why a 64-grid design can be folded at all mountain valley
circle-packingPacking the flaps One circle per flap, radius equal to that flap's length, packed into the square without overlapping. The packing determines the crease pattern; the leftover paper between circles becomes the structure that joins the flaps together. leg leg arm arm head the check closest approach 0.0000 no overlap — the packing is valid circles use 71% of the sheet the rest becomes the body efficiency is how much of the square the circles can claim, and it is an open problem the dashed skeleton is the subject; the circles are what it costs
curved-crease
deployablesWhat folding is used for Deployed area against packed area for several engineered folds. The pattern earns its place when something has to be large in use and small in transit, and every one of these is a case where nothing else would fit. Miura solar array 17× Space Flyer Unit, 1995 airbag folding 25× stored for years, opens in 30 ms heart stent 6× threaded through an artery starshade 11× 26 m disc, 2.5 m launch tube map fold 9× the original problem packed deployed the ratio is what is bought; one degree of freedom is what makes it reliable
developable
dividing-the-squareDividing a square into 3, exactly Haga's theorem. Folding one corner onto a point part-way along an opposite edge produces exact rational divisions elsewhere on the sheet — so a square can be divided into any whole number of parts by folding alone, with no measurement and no accumulated error. 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
doubling-the-cubeDoubling the cube in one fold Messer's construction. Dividing the sheet into thirds and folding one corner onto an edge divides that edge in the ratio one to the cube root of two — the classical problem Greek geometry could not solve, obtained from a single crease. 1 ∛2 ∛2 = 1.259921, and the point sits at 0.442493 one crease, for a problem Greek geometry could not solve
how-many-foldHow many assignments actually fold For a fixed set of crease lines, the number of mountain-and-valley assignments that satisfy the local conditions, against the number of assignments there are. The valid ones are a small and shrinking fraction, which is the quantitative form of the claim that flat-foldability is rare. one degree-4 vertex 4 of 16 4 creases · 25.0% survive one degree-6 vertex 8 of 64 6 creases · 12.5% survive the preliminary base 112 of 256 8 creases · 43.8% survive and these are only the local tests — a pattern can pass every vertex and still collide once the layers stack, which is the hard part
local-not-globalEvery vertex passes, which is not enough The local conditions are checked at each vertex independently, and a pattern can satisfy all of them and still fail to fold, because the layers have to stack without passing through one another. Deciding that for a general pattern is NP-hard, so no figure can settle it. 6 interior vertices, every one satisfying both theorems what the local tests see angles at each vertex assignment at each vertex what they cannot see whether layer 3 passes through layer 7 whether a flap has room to exist whether the order is consistent everywhere Bern and Hayes, 1996: NP-hard so this pattern is checked, not proved
maekawa-reason
miura-folded
miura-patternThe Miura fold A grid of identical parallelograms. The assignment is the whole trick: the horizontal creases alternate by row, and the vertical ones change assignment every time they cross a row, so each vertex ends up three of one and one of the other rather than two and two. at every vertex three of one, one of the other 15 interior vertices, all identical what the sheet gains one degree of freedom, not many it opens and closes in both directions at once a negative Poisson's ratio 22 mountain and 16 valley creases · 39.3 sheet-widths of folding mountain valley raw edge
Miura fold — sheet 170×106.83 mm — 22 mountain, 16 valley, 1049.4 mm of crease
one-vertex-motion
packing-ratio
preliminary-baseThe preliminary base Both diagonals and both midlines of a square, with the assignment that folds flat. Eight creases meet at the centre in equal sectors, so Kawasaki is satisfied by any assignment and Maekawa is the binding condition — five of one and three of the other, never four and four. at the centre 8 creases, all sectors 45° 3 mountain, 5 valley difference 2 — Maekawa holds four and four would fail, which is what most people draw fold every line, then collapse — the four corners meet mountain valley raw edge
Preliminary base — sheet 150×150 mm — 3 mountain, 5 valley, 724.26 mm of crease
rigid-versus-flatTwo different questions Flat-foldability asks whether a pattern can reach a flat state. Rigid-foldability asks whether it can get there without any face bending on the way. The second is much stronger, most patterns that satisfy the first fail it, and everything that gets manufactured lives in the smaller set. folds flat folds rigidly as panels in the outer set the bird base most traditional models anything with a squash fold in the inner set the Miura fold the Yoshimura pattern everything ever manufactured paper cheats by bending very slightly; sheet metal does not
square-twist
thickness
tree-to-base
trisection
vertex-conditionsA vertex that folds flat Four creases at one point, with the sectors between them measured and both flat-folding conditions evaluated. Kawasaki constrains the angles and Maekawa constrains the assignment; a vertex needs both, and they are independent of one another. V M M M 60° 90° 120° 90° Kawasaki 60° + 120° = 180° 90° + 90° = 180° both 180° — satisfied Maekawa 3 mountains, 1 valleys difference 2 exactly 2 — satisfied angles sum to 360° which is what a flat sheet requires mountain valley
What each toolset can reach The numbers each construction method can produce. A compass generates square roots and so reaches degrees that are powers of two; a fold solves cubics and reaches products of powers of two and three. The gap between the two rows contains the three classical impossible problems. straightedge alone rational no new numbers at all straightedge and compass degree 2^k √2, the regular 17-gon one fold at a time degree 2^a 3^b ∛2, the trisected angle, the regular 7-gon everything past here is out of reach of Euclid's tools doubling the cube, trisecting the angle and the regular heptagon all live in the gap
what-paper-is-not
why-a-circle
yoshimuraThe Yoshimura pattern The diamond pattern a thin-walled cylinder falls into when it is crushed axially. Nobody designed it — it is the buckling mode with the lowest energy, and it was named after the engineer who described it in aluminium tubes rather than after a folder. found in crushed drink cans, tree bark and deployable booms the pattern is a consequence of thin-wall buckling, not of a design mountain valley