Origami & Folding
what one sheet can do
Series

Crimping — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the waterbomb tessellation's odd vertexdegree six, and this site prints nine of them on one sheetevery assignment64passes all four conditions30has a flat folded state1812 labellings satisfy every condition the subject has and have no flat folded state

    Crimp it away and ask again

    Four conditions decide whether a vertex folds flat, and they decide it exactly at a vertex whose sectors are all different sizes. Everywhere else they over-count: two markings of every tied four-crease vertex, twelve of the degree-six vertex this site prints nine of on one sheet. What decides the case is not a fifth condition but a procedure — fold the smallest sector away and ask the smaller vertex.

    part 1 · flat-folding
  2. degreevertices visited per letteringcrimps needed48 of 16 fold32630 of 64 fold1038112 of 256 fold41410420 of 1024 fold2065121584 of 4096 fold12376The work grows by a factor of about 6.0 for every two creases added; the necessity grows by one.

    A tie is not a decision

    The crimp reduction decides a vertex by folding its smallest sector away, and where two sectors tie for smallest it has no forced move and must try each of them. That search is not rare — on the vertex at the centre of the first base anybody folds it happens for fourteen of the sixteen letterings — and it has never once changed the answer.

    part 2 · flat-folding

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