Found before it was designed — the buckling first generator
buckling-first is one function. Everything below was drawn by it when the site was
last published, at settings taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
With nothing specified
cols: 5, rows: 4, sq: 300
cols: 5, rows: 4
cols: 6, rows: 5
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- developability holds at every interior vertex — its sectors close to 360° — 14 checked ×5
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 14 checked ×5
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 14 checked ×5
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 14 checked ×5
- the pattern a crushed cylinder finds satisfies the flat-folding theorems at all 14 of its interior vertices — nobody chose them ×5
- every interior vertex has the same degree, 6 — a repeating unit rather than a drawing that happens to close ×1
- the three-to-one split holds at every vertex of the buckled pattern, which is Maekawa's theorem arriving from a shell rather than from a proof ×1
- the Yoshimura is put past all four theorems before it is drawn ×1
- the Yoshimura pattern is put past all four theorems before it is drawn ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
Almost every pattern fails
Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.
Found before it was designed
Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.
All the figures · The who found it, and when field · The patterns a reader can fold