Origami & Folding
what one sheet can do
Concept

Haga's theorem — where it appears

The result that folding a corner of a square to a point on the opposite edge produces exact rational divisions. It came from a schoolteacher's classroom rather than from a mathematics department.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

Dividing without measuring

A square can be divided into any whole number of equal parts by folding alone — exactly, with no ruler, and with no error to accumulate. The construction is one fold and a theorem nobody expected.

construction · Exact division the midpointwhat comes outcrease on the left edge 3/8crease on the right edge 7/8the folded edge crosses at2/3exact, and a trisectionthe corner is placed by folding, not by measuring — which is why theresult is exact

A schoolteacher's theorem

Kazuo Haga folded a corner of a square to the midpoint of the far side and found exact thirds. The construction needs one fold, no measurement and no compass, the numbers that come out are exactly 3/8, 7/8 and 2/3, and it was found by a biology teacher looking for something to do with a classroom.

history · Pedagogy 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.

construction · Paper proportion 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.

construction · Paper proportion 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.

construction · Paper proportion

Named alongside it

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

Exact divisionRational divisionReference pointConstructionPaper proportionError propagationGridOrigamicsPedagogySheet shape

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