Origami & Folding
what one sheet can do
Series

Reference points — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the bare sheet4 references · 4 linesnothing has been foldedafter 1 fold9 references · 12 lineshalves, and nothing elseafter 2 folds565 references · 92 lineshalves, thirds, fifths — and worsea fold is an alignment, and an alignment needs something already on the paper to align565 references after 2 folds, and the count is finite however many folds are allowed

    A fold needs something to align

    Every axiom names things that must already be on the paper — a point to fold onto a point, a line to bring to a line. So what a folder can build is bounded by what they can refer to, and that set is finite at every depth: nine references after one fold, several hundred after two, and every one of them computable in advance.

    part 1 · construction
  2. the bare sheet4 references · 4 linesnothing has been foldedafter 1 fold9 references · 12 lineshalves, and nothing elseafter 2 folds565 references · 92 lineshalves, thirds, fifths — and worsea fold is an alignment, and an alignment needs something already on the paper to align565 references after 2 folds, and the count is finite however many folds are allowed

    Cheap where it reaches

    Two folds from a bare square put marks at a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth — and at no seventh, ninth or eleventh at all. A rule that reaches every fraction takes n folds to reach one nth. The systematic route and the short one disagree everywhere, and neither of them knows about the other.

    part 2 · construction
  3. what each round of folding reaches, and how close together it ison a sheet 150 mm squaremarksclosest pairmedian gapwithin 0.2 mmone fold, every axiom975.000 mm75.000 mm0.0%two folds, every axiom5650.520 mm2.700 mm0.0%three folds, point onto point only5538230.000 mm0.058 mm94.4%a crease in ordinary paper is about that wide, so the last column is the share of marks a folder cannot separate

    Closer than a crease is wide

    One fold from a bare square leaves nine marks, seventy-five millimetres apart. Two folds leave five hundred and sixty-five, the closest pair half a millimetre apart. Three folds — using one axiom of the seven — leave half a million, and ninety-four per cent of them have another mark within a fifth of a millimetre. What bounds a folder is not what the axioms reach; it is what the paper can tell apart.

    part 3 · construction
  4. 2 folds from a bare sheet, every proportion at the same areamarks separated by at least 0.3 mm on a 150 mm sheetthe squarewhat origami paper is sold as565 marks · 92 distinct foldsthe A serieshalves into itself45,705 marks · 752 distinct foldstwo squaresa square cut the long way26,155 marks · 540 distinct foldsthe 1 : √3 rectanglethirds into itself42,746 marks · 732 distinct foldsthe golden rectanglenot in the halving family43,233 marks · 732 distinct folds

    The sheet decides which points exist

    Every measurement of what folding can locate has been made on a square, because origami paper is sold square. Hold the area fixed and change the proportion: one fold reaches nine marks on a square and twenty-nine on the A-series rectangle, and two folds reach 565 against 45,705. The square is the worst of five proportions at both depths, and the reason is its own symmetry.

    part 4 · construction
  5. one round of folds through two points and folds placing one point on anothera crossing that lands inside the hole is not a reference and is not drawnwith a hole: 212 referencessolid: 9from 8 corners and 8 edgesfrom 4 corners and 4 edges

    A hole is an edge

    A folder's first fold has to be specified by aligning things that are already there, and what is already there is the sheet's outline. Cut a square hole in the middle and the outline doubles: one round of alignments reaches nine references on a plain square and two hundred and twelve on a holed one — more than the plain square reaches in two rounds.

    part 5 · construction
  6. the bar is the share of each round's new references that lie on the sheet's own edgeafter one fold80%4 on the edge · 1 inside · 12 fold linesafter 2 folds9%48 on the edge · 508 inside · 92 fold linesan edge is a line a fold can cross twice, so it yields marks with the folds and not with their pairs

    The edge was there first

    A folder's first fold has nothing to align to but the sheet's own outline, and it shows in where the marks land: four of the five references the first fold adds are on the paper's edge. By the second fold the edge holds forty-eight of five hundred and fifty-six new ones. The rim is where references are cheap and it fills up, because an edge is a line a fold can cross twice while two folds inside the paper cross once each.

    part 6 · construction
  7. 565 references · worst gap 0.089 of a sheet · under 0.092the disc is where a folder has the least to align againstthe faint lines are the folds themselves; a reference is where two of them cross on the paper

    How far from the nearest reference

    One fold puts nine reference points on a square sheet and two folds put five hundred and sixty-five. That is sixty-three times as many points, and it brings the worst-covered spot on the paper from a third of a sheet away to a twelfth — four times closer. A count of references is not a measure of what a fold buys, because a set of points can be arbitrarily crowded and still leave most of the sheet out of reach.

    part 7 · construction
  8. the bar is the largest gap between anywhere on the sheet and a referencea third fold specifies more folds than can be listed, so a sample of them is taken insteadnone of them0.089565 references, from two folds50 of them0.0763,498 references · 0.02% of the round100 of them0.0508,056 references · 0.04% of the round200 of them0.04723,480 references · 0.07% of the round400 of them0.02274,694 references · 0.15% of the round800 of them0.014270,882 references · 0.29% of the roundevery row is a lower bound on what the whole round would buy, because leaving folds out can only make the gap larger

    The third fold cannot be listed

    Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.

    part 8 · construction
  9. the bar is the share of each round's new references that lie on the sheet's own edgeafter one fold80%4 on the edge · 1 inside · 12 fold linesafter 2 folds9%48 on the edge · 508 inside · 92 fold linesan edge is a line a fold can cross twice, so it yields marks with the folds and not with their pairs

    A reference on a sheet with no corner

    Every construction in this subject begins from the sheet's own boundary: two edges meet at a corner, a corner is a point, and a point is what an axiom takes as input. A cylinder has two circles of edge and no corners at all, so a construction on one has nothing to start from and the seam is not a mark.

    part 9 · construction

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