Origami & Folding
what one sheet can do
Series

Colour change — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

    Bringing the other side to the front

    Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

    part 1 · design
  2. patternfootprint with both colours over itlayers thereThe preliminary base8 panels, 4 one way up and 4 the other100%8.0The Miura fold24 panels, 12 one way up and 12 the other100%9.2The square twist9 panels, 5 one way up and 4 the other65%4.1The hexagon twist13 panels, 7 one way up and 6 the other72%4.1The Yoshimura pattern65 panels, 32 one way up and 33 the other100%60.0Fold and cut — the triangle7 panels, 4 one way up and 3 the other4%6.4The tapered corrugation28 panels, 14 one way up and 14 the other100%8.4The waterbomb tessellation52 panels, 26 one way up and 26 the other100%31.1

    Decided before the design

    A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.

    part 2 · design
  3. the bar is the share of the footprint showing the side that started face upthe rest of it shows the other side, and neither is chosen by anybodyThe preliminary base0.3%2 of 8 panels in viewThe square twist47.4%7 of 9 panels in viewThe hexagon twist90.8%10 of 13 panels in viewFold and cut — the triangle100.0%1 of 7 panels in view · 2 statesa panel out of view from above is not hidden — it is under the pile, and turning the sheet over shows a different set

    Which side arrives

    A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.

    part 3 · design
  4. the bar is the footprint, split by how deep the other side liesone layer downtwo layers downnot under this point at allThe preliminary base8 panels · 1 stateThe square twist9 panels · 1 stateThe hexagon twist13 panels · 1 stateFold and cut — the triangle7 panels · 2 statesa 2 × 2 Miura patch4 panels · 1 statea 3 × 2 Miura patch6 panels · 3 statesa 3 × 3 Miura patch9 panels · 6 statesa 4 × 3 Miura patch12 panels · 11 statesthe letter fold3 panels · 2 statescounted over every folded state and from both faces, so nothing here is one lucky pile

    One sheet down

    A colour change has been priced by how deep the pile is — eight layers over every point of the preliminary base, six on a small Miura. Ordered, the piles say something else: wherever the other side of the paper lies under a point, it is the next sheet down on every pattern with one folded state, and never more than two down on any. And relettering the same creases cannot reach it. Of 112 letterings of the preliminary base that fold, every one shows either the printed face or the whole face turned the other colour; the square twist's eight only turn its face round.

    part 4 · design
  5. one crease 0.3 along each edge from a corner of the preliminary base: every face its letterings showthe side that started face upthe other sidenothing changed0% · 0 letterswhere it lands18% · 2 letterswhere the corner was18% · 2 lettersboth of those36% · 2 lettersthe rest64% · 3 lettersthe rest and where it lands82% · 3 lettersthe rest and where it was82% · 3 letterseverything100% · 2 letters

    A corner turned shows the sheet beneath

    Relettering the preliminary base turns its whole face one colour or the other and nothing between. One added crease is the cheapest change that does better, and the angle condition leaves it a single direction: across a diagonal at right angles, cutting a corner off. Placed s along each edge from the corner, it divides the face into three regions — where the corner lay and where it lands, each twice s squared of the face, and the rest — and the letterings of the result colour those three regions in all eight possible ways.

    part 5 · design
  6. two opposite corners cut 0.3 and 0.15 from their tips: every colouring of the five regions, from aboveunder each face, its colouring region by region in the order of the key, and the fewest of the base's own letters changed1 where both corners lay · 2 where the small corner lands · 3 the band only the large corner left · 4 where the large corner lands · 5 the rest of the facethe side that started face upthe other side0 letters3 letters2 letters3 letters2 letters2 letters3 letters4 letters3 letters5 letters4 letters3 letters5 letters4 letters3 letters2 letters3 letters5 letters4 letters3 letters5 letters4 letters3 letters5 letters4 letters2 letters3 letters3 letters5 letters3 letters2 letters2 letters

    Two corners compete across an edge

    One crease across a corner of the preliminary base cuts its face into three regions, and the letterings colour them all eight ways. Two creases across two corners cut it into five, and the question was whether the corners compete for the paper beneath. It depends on which two. Opposite corners colour the five regions in all thirty-two ways on either face. Adjacent corners, joined by an edge the base folds in half, reach twenty-nine on each face and lose three; the three one face cannot show are the other face's three with every side exchanged. And two corners cut at the same distance land on each other, so the face is back to three regions however many states the letters make.

    part 6 · design

All series