Origami & Folding
what one sheet can do
Concept

Reference points — where it appears

The marks a folder can locate on a sheet before making the fold they want. Every construction needs something to align to, so what a fixed set of operations reaches from a bare sheet — and how crowded those marks are on real paper — bounds what can be folded exactly.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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.

construction · Reference points 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.

construction · Reference points 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.

construction · Reference points 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.

construction · Reference points 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.

construction · Reference points

Named alongside it

The objects these essays reach for when they reach for this one.

Sheet shapeConstructionThe Huzita–Hatori axiomsReference pointAxiomAxiomsBoundaryBoundary vertexClosureConstructibilityCrease patternCrease radius

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