Origami & Folding
what one sheet can do
Series

Paper proportion — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. proportion 1.3two shapes, in turn1.3001.5381.3001.5381.300proportion √2 = 1.4142one shape, throughout1.4141.4141.4141.4141.414halving turns a proportion of r into one of 2/r, and those are the same number only at √2the 1.3 sheet is a different shape after every fold; the √2 sheet is the same shape after all of thema square metre at √2 is 840.9 × 1189.2 mm, which is the 841 × 1189 printed on a sheet of A0

    The rectangle that keeps its shape

    Halving a rectangle across its long side turns a proportion of r into one of 2/r, so almost every sheet comes out of the fold a different shape from the one that went in. Exactly one does not, and it is not a shape anybody chose.

    part 1 · construction
  2. 1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to build1 : √6 = 2.4495 → 6 parts, 5 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes

    One member of a family

    A4 halves into A5 and keeps its shape, which is the one thing everybody knows about paper sizes. The property is not about halving and not about two: a rectangle in the ratio √n divides into n copies of itself, for every n, and every one of those rectangles can be folded out of a square one diagonal at a time.

    part 2 · construction
  3. 1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes

    The proportion a band asks for

    √2 is a shape: a rectangle either has it or does not, and what it buys is that halving returns the same shape. √3 is what a Möbius band needs, and it is a different kind of number — a minimum rather than a shape, with every longer strip working and no shorter one.

    part 3 · construction
  4. the same fold, on five sheetsa corner brought to the midpoint of the far edge, and what comes outsheetleft edgeright edgethe crossingwhat happenedsquare1.000 × 1.0003/87/82/3a third, exactlyA series, tall1.000 × 1.4147/1611/162/7a number, and not a thirdA series, wide1.414 × 1.000—1.2500—the crease leaves the paper3 : 2, tall1.000 × 1.5004/92/31/4a number, and not a third3 : 2, wide1.500 × 1.000—1.3438—the crease leaves the paper2 sheets answer and are wrong; 2 refuse — and the difference between the two is a right anglethe alignment does not know what shape the paper is, and neither does the folder following it

    A construction assumes its sheet

    Haga's fold gives exactly two thirds on a square. Run the same alignment on an A-series sheet held tall and it gives exactly two sevenths, with the crease meeting the vertical edges at seven sixteenths and eleven sixteenths — every one of them a clean fraction, none of them what the recipe promised. Turn the same rectangle through a right angle and the crease leaves the paper instead, which is the loud failure rather than the quiet one.

    part 4 · construction
  5. seven constructions on five sheetswhere each construction's point lands, as a fraction of the sheet, against the squaresquareA series, tallA series, wide3 : 2, talldouble, widehalve it, edge onto edgea fold along the stretch1/2, 0the samethe samethe samethe samea third, from two crossing linescrossings only1/3, 1/3the samethe samethe samethe samea fifth, by repeated crossingscrossings only1, 1/5the samethe samethe samethe samea third, by Fujimoto's halvingsfolds along the stretch0.333, 0the samethe samethe samethe sameHaga's fold, corner to midpointa fold across a slant1, 2/31, 2/7off the paper1, 1/4off the papera corner halved, edge onto edgea fold across a slant1, 11, 0.7070.707, 11, 2/31/2, 1a corner onto the opposite cornera fold across a slant1, 0off the paper3/4, 0off the paper5/8, 0a stretch along the edges keeps crossings, midpoints and folds along the edges; it does not keep a fold across a slant

    A stretch keeps crossings

    A rectangle is a square stretched along its edges, and a stretch along the edges keeps straight lines straight, crossings as crossings, midpoints as midpoints and the fraction a point divides a segment into. So a construction made only of those — halve an edge, cross two lines — lands at the same fraction of every rectangle, and four standard constructions do. A fold across a slanted line is a reflection the stretch does not keep, and every construction that uses one — Haga's, a corner halved, a corner brought to its opposite — returns a different point on some rectangle, or none.

    part 5 · construction
  6. creases to mark each fraction of an edgeportable: every rectanglesquare onlyA-series sheet only0123456creases11112112313124131234515/2/3/4/5/6/7/8/9/10/11/12the portable column is one number for every rectangle; each sheet's column is true of that sheet alone

    What the square saves

    A construction made only of crossings, midpoints and folds along the edges lands at the same fraction of every rectangle, so its cost is one number for all of them. Counted crease by crease against every fold the first four axioms allow, that portability costs about a crease and three quarters a fraction up to twelfths — and the square is not the cheapest sheet to give it up for. An A-series sheet, where Haga's fold goes silently wrong, marks two sevenths in two creases; the square needs three, and any sheet at all needs five.

    part 6 · construction

All series