Where the cubic comes from

That folding solves cubics is usually presented as a property of the sixth axiom, and the presentation is complete but unsatisfying: the axiom is stated, the algebra is done, a cubic appears. It answers whether and not why, and the why turns out to connect the axiom list to a piece of nineteenth-century graphics.

The why is a construction from 1867 that has nothing to do with paper, and once it is in hand the fold stops being a special fact about folding and becomes an instance of something general.

Lill’s construction #

Take a polynomial and lay its coefficients out as a path. Start at the origin, walk a distance equal to the leading coefficient, turn a right angle, walk the next coefficient, turn again, and so on; negative coefficients are walked backwards.

Now send a ray out from the origin. Let it bounce off each segment of the path — off the line the segment lies on, at a right angle each time — and see where it ends up. For most launch angles it misses the path’s endpoint. For particular ones it arrives exactly, and for those,

$$x = -\tan\theta$$

is a root of the polynomial.

The proof is a page of similar triangles and is worth doing once. Each bounce multiplies a length by $\tan\theta$ and subtracts the next coefficient, which is Horner’s scheme — the way a polynomial is evaluated efficiently — carried out in geometry. Arriving at the endpoint means the final remainder is zero, which is what it means for $x$ to be a root.

That last sentence is the whole thing. Lill’s path is a polynomial evaluator, and the ray is the argument.

Working the path by hand #

The construction is worth carrying out once on a specific cubic, because it is short and because doing it removes any suspicion that the picture is doing work the argument is not.

Take $8x^3 + 4x^2 - 4x - 1$, whose roots are the cosines of $2\pi/7$, $4\pi/7$ and $6\pi/7$ — the cubic that stands between a compass and a regular heptagon. Lay the legs out: eight to the right, four up, four to the left (because the coefficient is negative and the turn has already been made), one down.

Launch a ray at angle $\theta$ and follow it. It meets the second leg’s line at height $8\tan\theta$. Turning a right angle and running to the third leg’s line uses a horizontal distance of $\tan\theta\,(4 - 8\tan\theta)$. Turning again and running to the fourth uses $\tan\theta$ times what is left. Requiring the ray to finish exactly on the path’s endpoint gives

$$8u^3 - 4u^2 - 4u + 1 = 0, \qquad u = \tan\theta,$$

which is the original cubic with $x = -u$. No geometry beyond right angles has been used, and the polynomial has appeared out of three multiplications.

That is Horner’s scheme. Evaluating $a_3x^3 + a_2x^2 + a_1x + a_0$ as $((a_3x + a_2)x + a_1)x + a_0$ is three multiply-and-adds, and each bounce of the ray performs one of them: multiply the running value by $\tan\theta$, subtract the next coefficient. Landing on the endpoint is the final remainder coming out at zero.

What a fold does #

A fold is a reflection. Folding so that a point $P$ lands on a line $\ell$ means folding along a line every point of which is equidistant from $P$ and $\ell$, and the set of such points is a parabola with focus $P$ and directrix $\ell$. So a fold that brings a point onto a line is a tangent to a parabola.

The sixth axiom asks for a single fold that brings $P_1$ onto $\ell_1$ and simultaneously $P_2$ onto $\ell_2$. That is a line tangent to two parabolas at once — a common tangent — and two conics have three common tangents in general.

Three common tangents, three roots. The cubic is not smuggled in by the algebra; it is the degree of the system, and it was there before anybody wrote an equation down.

Why three tangents and not four #

“Two conics have three common tangents in general” is the step the whole essay turns on, and it is not the general fact. Two conics have four, and where the fourth one goes is the reason folding reaches degree three rather than degree four.

The clean way to see it is in the dual plane, where a line of the ordinary plane is a point and a conic becomes another conic. A line tangent to both curves is a point lying on both dual conics, and two conics meet in four points. So four common tangents, for two conics in general position.

Parabolas are not in general position. A parabola is exactly the conic that is tangent to the line at infinity, so both dual curves pass through the one point that represents that line, and one of the four intersections is spoken for before anything else is decided.

Three intersections remain, and the line at infinity is not a crease anybody can make. So a fold that brings two points onto two lines has three solutions rather than four, and the system it solves is a cubic.

Which says the degree was fixed by reflection #

That is the why the essay set out to find, and it lands somewhere better than the algebra does.

The degree of the sixth axiom is not a property of paper, of alignment, or of how many conditions a crease can carry. It is a consequence of one fact about reflection: a fold that carries a point to a line traces the set of points equidistant from both, and that set is a parabola. Had the operation produced ellipses, two of them would have four common tangents and a single fold would solve quartics.

It also says the degree cannot be improved by cleverness. Any construction that specifies a crease by two point-to-line alignments is asking for a common tangent to two parabolas, and the line at infinity takes one of the four every time.

The discriminant, and how many rays get drawn #

The same count has a cheap check attached, and every path drawn here passes it.

A cubic has three real roots exactly when its discriminant is positive, and the number of rays that arrive at the end of Lill’s path is the number of real roots. For $8x^3+4x^2-4x-1$ the discriminant is 3,136; for $x^3-x^2-2x+1$ it is 49. Both positive, so three rays each, which is what the figures draw.

The other paths in this essay are drawn from the same rule and can be read off it: $x^3-3x+1$ has discriminant 81 and arrives three ways, while $x^3-2$ and $x^3+x+1$ have discriminants $-108$ and $-31$ and arrive once each. The first two numbers here are also perfect squares — $56^2$ and $7^2$ — and a cubic whose discriminant is a square has a cyclic rather than a full symmetry group. That is not a coincidence of the choices: the first cubic is the one the regular heptagon needs, and the second is the same field written at a different scale. Two examples, one number field, which is worth knowing before reading the second figure as an independent case.

The two are the same picture #

Now put the constructions side by side. Lill’s ray bounces at right angles off a sequence of lines. A fold reflects the plane across a line. The bouncing ray is the path a common tangent traces through Lill’s rectangle, and the two right-angled turns of the ray at the first and last bounce are exactly the two point-onto-line conditions the sixth axiom imposes.

Margherita Beloch made the connection in 1936 and it is the reason her name is on the fold. She observed that Lill’s construction reduces root-finding to producing a particular bouncing path, that a single fold produces exactly such a path, and therefore that paper folding solves cubics. The sixth axiom is Lill’s method performed by reflection.

Reading it that way explains three things that are otherwise separate facts.

It explains why three roots. The bouncing path can leave the origin at three angles that arrive, because the cubic has three roots, because two conics have three common tangents. All three statements are the same statement.

It explains why the axiom list stops where it does. One fold imposes at most two alignment conditions, two conditions give a common tangent to two conics, and a common tangent gives a cubic. A construction that solved a quartic would need a fold satisfying three conditions, and a fold does not have three degrees of freedom to spend.

And it explains why straightedge and compass cannot. Those tools intersect lines and circles, which is a system of degree two; every construction is a tower of square roots. Folding intersects a line with two parabolas, which is degree three, and adds cube roots to the tower. The heptagon needs one; so does doubling the cube.

What the tower looks like #

The algebraic account is worth stating alongside the geometric one, because the two answer different questions and readers usually want both.

A number is constructible with straightedge and compass exactly when it lies at the top of a tower of field extensions each of degree two — every step is the solution of a quadratic, which is what intersecting a line with a circle produces — the same bound that decides which regular polygons a compass reaches. The degree of such a number over the rationals is therefore a power of two, and anything whose minimal polynomial has degree three is out of reach. That is the whole of the classical impossibility argument, and it disposes of trisection, cube-doubling and the heptagon in one line each.

Folding adds one step type to the tower: a degree-three extension, from the sixth axiom. The reachable numbers are then those at the top of a tower whose steps are of degree two or three, which is a strictly larger set — and it is exactly the set the geometric argument predicts, since three common tangents is a cubic and nothing above.

The two accounts are not alternatives. The field theory says what the reachable set is; Lill and Beloch say why the operation has that degree, which is the question the field theory takes as given.

Which theorem was checked, and how #

Both figures compute rather than illustrate, and both assert the property that would make them wrong.

Lill’s figure computes the real roots of the cubic by bracketing and two hundred rounds of bisection, then constructs the bouncing ray for each root and measures how far it lands from the end of the path. A miss of more than a billionth throws and stops the build. The number is printed on the figure, so a reader is not asked to take the assertion on trust.

Beloch’s figure does the same with the tangency: for each root it computes where the fold line touches each parabola and requires the two to agree exactly. That check has found nothing yet, and it is what makes the picture evidence rather than decoration.

The root-finder is shared between the two figures, which matters. Each once had its own copy of the same bracketing loop; a change to one would have left the other alone, and the two would have quietly disagreed about which roots there were.

What the picture cannot show #

Lill’s method is drawn here for a cubic, and the construction works for any degree — a quintic is five legs and four bounces. What does not carry over is the folding: one fold produces two bounces, not four, so the correspondence between Lill and paper is specific to the cubic.

The figure also draws only real roots. A cubic with one real root has one launch angle that arrives, and the other two roots are complex and have no bouncing path at all. Nothing in the picture indicates that they exist, and the number of drawn rays is not the number of roots.

And a fold is drawn as a line. In the hand the fold is made by moving the paper until two alignments hold at once, which is a physical search with two conditions and one degree of freedom — a slightly uncomfortable operation that folders call a neusis fold, and which is why the sixth axiom is the one that feels different to perform.

The idealisation underneath #

Every construction on this rung assumes that alignment is exact: that a point can be brought onto a line with no error, and that a crease is a line.

The consequence is specific rather than general. A quadratic construction with a small alignment error gives an answer with a comparable error. A cubic construction can do worse, because the common tangent is a solution to a system whose conditioning depends on the configuration — two parabolas that meet at a shallow angle have a tangent whose position is very sensitive to where their foci are. So the sixth axiom is the least forgiving of the seven in practice, and this is a fact about the algebra rather than about anybody’s hands. It is a different failure from the accumulating error of an iterative division, where the arithmetic is forgiving and the folding is not.

Why the sixth axiom feels different in the hand #

Anybody who has performed the seven operations notices that six of them are placements and one is a search, and the account above says why.

Five of the axioms specify a fold by conditions that pin it directly: a crease through two points, a crease bisecting two points, a crease bringing one line onto another. The fold line is found by an alignment that either holds or does not, and the paper snaps into place.

The sixth asks for two simultaneous alignments and gives only one crease to satisfy them with. The paper is slid until the first point sits on its line, which can be done at a continuum of positions, and among those positions there are up to three at which the second alignment also holds. Finding one means moving the paper along the family and watching the second condition approach.

The classical name for an operation with this shape is neusis — a marked-ruler construction in which a length is slid until two conditions hold. The Greeks knew that neusis solves problems the compass cannot, used it freely, and treated it as outside the canon; Archimedes trisects an angle with one. Folding is a neusis device that happens to be made of paper, which is also how angle trisection escapes the classical impossibility, which is a more accurate description of the sixth axiom than any list of alignments.

The surprising connection #

Lill was an Austrian artillery officer, and his construction was published in 1867 in a journal of mathematics as a graphical method for finding roots — one of many such methods from a century when computing a root meant drawing it.

Nearly every other graphical method from that era is forgotten, because a calculator does the job better. This one survived by being about something else. It is the only one that turned out to describe a physical operation, and the operation is folding a sheet of paper.

There is a second, quieter connection. The bouncing ray is Horner’s scheme, and Horner’s scheme is how a computer evaluates a polynomial today — the same sequence of multiply-and-add, in a register file rather than on a rectangle. So a fold, a nineteenth-century drawing method and the inner loop of a numerical library are three renderings of one algorithm, and the fold is the oldest of them by several thousand years, having been performed by people who were not doing arithmetic.

Who found it, and when #

Eduard Lill published the construction in 1867. Margherita Piazzolla Beloch, working in Ferrara, connected it to paper folding in 1936, and the fold that produces a common tangent to two parabolas is named for her. Her work went largely unnoticed outside Italy for decades.

Humiaki Huzita’s axiom list of 1991 includes the operation as the sixth axiom without reference to Beloch; Koshiro Hatori found the seventh in 2001, and Jacques Justin had the complete list in 1989 in a paper nobody read at the time. Robert Lang proved in 2003 that the list of seven is complete, and Thomas Hull and Roger Alperin set out the field-theoretic account that connects the axioms to the tower of extensions.

The historiography is a small lesson in itself: the key result was published twice, sixty years apart, in languages and venues that the eventual audience did not read.

The ladder from here #

This rung explains the sixth axiom rather than stating it. Below it, folding beats the compass establishes that it does; why the list stops at seven establishes that there is nothing further of the same kind.

Above it the question is what happens with more than one fold at a time. Two simultaneous folds impose four conditions and reach quintics; the multifold axioms of Alperin and Lang generalise the seven, and each additional simultaneous fold raises the degree. That ladder runs a long way and nobody has folded anything at the top of it.