Origami & Folding
what one sheet can do
Series

As drawn — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

    The vertex the list does not have

    Every condition this collection checks is asked at a vertex of a crease pattern, and a crease pattern is handed to the checker as a list of points and segments. A reader is handed ink. Read the same patterns the second way and eight printed sheets gain nothing at all — while four tessellation patches gain 12, 18, 12 and 5 vertices that nobody wrote down, every one of them a place where two creases were drawn across each other.

    part 1 · flat-folding
  2. the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge

    Two creases that cross

    A crossing is four creases at a point, so the four conditions of the subject apply to it — and three of them can be satisfied. It is developable at every angle, it satisfies the big-little-big lemma whenever its two lines carry different letters, and it satisfies Kawasaki's condition when the lines meet squarely. Maekawa's refuses it always, at every angle and under every lettering, because a crossing's four spokes belong to two creases and can only be four and none, two and two, or none and four.

    part 2 · flat-folding
  3. the bar is how many creases the found lettering writes differentlymeasured against the lettering the pattern's own construction producedthe square patch4545 of 84 creases · 31 of them buriedthe elongated patch6666 of 106 creases · 42 of them buriedthe hexagonal patch6767 of 142 creases · 45 of them buriedthe triangular patch8787 of 142 creases · 65 of them buriedthe rhombille patch155155 of 282 creases · 117 of them burieda buried crease has an interior vertex at each end, and no legal move ever changes one

    One solution of a search nobody ran

    A crease pattern arrives with its letters already on it, and they look like part of the drawing. They are not. Every construction here ends in a propagation, a propagation ends wherever its first guess took it, and the lettering that comes out differs from the one a search finds on between a half and three-fifths of the creases — on patterns whose own letters are perfectly good.

    part 3 · flat-folding
  4. each circle holds a crease shorter than a thousandth of the sheet12 fragments7.9·10⁻⁶ of a sheet · M5.9·10⁻⁵ of a sheet · M7.9·10⁻⁶ of a sheet · M5.9·10⁻⁵ of a sheet · V7.9·10⁻⁶ of a sheet · V5.9·10⁻⁵ of a sheet · Mand 6 more1018× to draw the longest142 creases and 60 interior vertices, 12 of the first and six of the second invisible

    The crease the drawing cannot show

    Twelve creases on a printed crease pattern are eight millionths of a sheet long. They are in every count the collection takes of that patch, they pass every theorem, and no printer resolves them and no hand folds them. They are also the only thing holding the folded sheet together.

    part 4 · flat-folding
  5. one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

    The drawing does not say what is glued

    One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.

    part 5 · flat-folding

All series