Origami & Folding
what one sheet can do
Series

Pedagogy — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

    The kindergarten was a geometry class

    Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.

    part 1 · history
  2. the midpointwhat comes outcrease on the left edge 3/8crease on the right edge 7/8the folded edge crosses at2/3exact, and a trisectionthe corner is placed by folding, not by measuring — which is why theresult is exact

    A schoolteacher's theorem

    Kazuo Haga folded a corner of a square to the midpoint of the far side and found exact thirds. The construction needs one fold, no measurement and no compass, the numbers that come out are exactly 3/8, 7/8 and 2/3, and it was found by a biology teacher looking for something to do with a classroom.

    part 2 · history
  3. the bar is the letterings with a folded statethe row is how many times the letter changes going round the central polygonthe ring reads as one letter032 pass every vertex · 28 have no order2 changes round the ring8192 pass every vertex · 184 have no order4 changes round the ring032 pass every vertex · 32 have no ordera twist looks like a twist when the ring reads as one letter, which is why this was never checked

    Taught with a wrong reason

    Four mountains and four valleys is what the preliminary base's symmetry suggests and Maekawa forbids it; a twist looks like a twist when its central ring reads as one letter, and no such lettering folds; a tessellation is verified because its unit is, and a forty-nine-panel patch of one had no folded state at all. In each case the conclusion taught is right and the reason offered for it is not, and the site that repeats them is this one.

    part 3 · history
  4. 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

    The reader decides the junction

    Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

    part 4 · history
  5. each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsa loop of 8 panels — no order existsthe arrows are the whole of the test — nothing here asks which panels lie over which

    The first thing about layers

    A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.

    part 5 · history
  6. the 16 repeating rules that fold, written outrows first, then the two column classes — and every one of them alternates down the columnrows · columns above|below20VV · MV|MV21MV · MV|MV22VM · MV|MV23MM · MV|MV24VV · VM|MV25MV · VM|MV26VM · VM|MV27MM · VM|MV36VV · MV|VM37MV · MV|VM38VM · MV|VM39MM · MV|VM40VV · VM|VM41MV · VM|VM42VM · VM|VM43MM · VM|VMfour ways of writing the rows times four ways of alternating the columns is sixteen, and there is nothing else

    Half the recipe is decoration

    Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.

    part 6 · history
  7. the rules that fold, against the clauses they all obeya clause says an odd or even number of some chosen letters are mountainsfamilyrulesfoldparity clausesthey allowa same-or-differ recipethe Miura fold6416216yesthe tapered leaf6416216yesthe Yoshimura pattern6426132nothe waterbomb tessellation51232364noa recipe of same and differ clauses allows exactly the rules its clauses allow; that is the test

    A recipe needs degree four

    The Miura's letters are taught as a recipe of same-and-differ clauses, and the recipe is exact: the sixteen repeating rules that fold are precisely the rules two clauses allow. The Yoshimura's twenty-six cannot have such a recipe, because twenty-six is not a power of two. The waterbomb's thirty-two is a power of two and still has none — its clauses allow sixty-four rules and half of them fail. The difference is one vertex: at degree four the counting theorem leaves a parity, and at degree six it leaves 'not all alike', which no clause of that kind can say.

    part 7 · history
  8. the clauses a recipe needs, family by familyeach recipe is checked to allow exactly the rules that fold, and nothing elsefamilyrulesfoldparitiesprohibitionsdegree-six kindsclauses in allthe Miura fold64162002the Yoshimura pattern64261223the waterbomb tessellation512323245a parity says an even or odd number of some letters are mountains; a prohibition says some letters are not all alike

    Two sentences for the Yoshimura

    No recipe made only of same-and-differ clauses picks out the Yoshimura pattern's twenty-six folding rules, because its vertices have six creases. A recipe allowed one other kind of sentence does, and it is short: an even number of the four zigzag classes are mountains, and no course carries the letter all four zigzags share. That is three clauses, one parity and a prohibition for each of the pattern's two kinds of vertex, and it allows exactly the twenty-six. The waterbomb tessellation, with four kinds of six-crease vertex, needs three parities and only two prohibitions, because its parities do half the prohibiting.

    part 8 · history

All series