Exact division — where it appears
Named by 13 essays across 2 fields — each of them below, with the objects they name alongside it.
The kindergarten was a geometry class
Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.
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.
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.
One crossing, and then another
Folding a strip into thirds by Fujimoto's method halves the error at every fold and never reaches a third. There is a construction that arrives instead: cross the square's diagonal with a line through the mark you already have, and the crossing lands on the next fraction exactly — one fold per step, all the way down.
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.
Exact is not accurate
This site has two ways of dividing a strip into equal parts: a ladder that lands on the fraction as a rational number, and Fujimoto's method, which never arrives. Read as mathematics that settles it. Read as instructions for somebody with a sheet of paper it settles nothing, and past four parts the method that never arrives is the one whose crease lands nearer the mark.
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.
The grid a division makes
Dividing a square into thirds in both directions is a construction: four creases, each exact, each landing on a rational the ladder can name. The object it leaves behind is a three-by-three map of stamps, and how many ways that folds is the oldest open problem in the subject — 1,368 at three, 300,608 at four, and unknown at five.
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.
Dividing a loop into n
Fujimoto's method divides a strip into any number of equal parts by folding badly and then folding the error in half, over and over. It converges because each step halves what is left over. On a closed loop there is no edge for the leftover to sit against, and what replaces the edge is the loop's own closure.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Rational divisionReference pointConstructionHaga's theoremPaper proportionConvergenceFujimoto's methodThe Huzita–Hatori axiomsPedagogySheet shapeThe axiomsApproximation