The heptagon a compass cannot reach

The regular heptagon is the first polygon that stops a compass. Triangle, square, pentagon and hexagon are all elementary; the seven-sided one is not merely hard but impossible, and it has been known to be impossible since 1837.

It takes one fold.

The question is about one number #

Constructibility looks like a question about geometry and turns out to be a question about arithmetic, which is the whole content of the classical results.

To build a regular n-gon is to divide a circle into n equal parts, which is to construct the angle 2π/n, which is to construct the number cos 2π/n. So the geometric question becomes: is that particular cosine reachable from the rationals using the operations the tool provides?

The compass provides square roots. Every intersection it can compute is the solution of a quadratic, so every number it can reach lies in a tower of extensions each of degree two, and the degree of any constructible number over the rationals is a power of two.

A fold provides cube roots as well, because axiom six solves a cubic. So the reachable degrees are products of powers of two and powers of three.

That is the whole difference, and everything else is working out which cosines have which degree.

It is worth being careful about what “reachable” means here, because the word does two jobs. A number is constructible when it can be written using the four arithmetic operations and the roots the tool supplies, starting from the numbers already on the sheet — which for a square of paper means the rationals. It does not mean that anybody has written down the expression, or that the expression is short, or that the fold sequence realising it is one a person would enjoy. Constructibility is a statement about a field, and a field is a set of numbers rather than a set of instructions.

The distinction matters because the classical impossibility results are often heard as claims about difficulty. They are not. Trisecting a general angle with a compass is not hard; it is outside the field, in the same way that the square root of two is outside the rationals. No amount of ingenuity crosses that boundary, and the boundary moves only when the tool changes.

The test, and where it comes from #

The degree of cos 2π/n over the rationals is φ(n)/2, where φ counts the integers below n sharing no factor with it. The derivation is a page of Galois theory and the result is a single line, so the test states cleanly:

Compass and straightedge reach the regular n-gon exactly when φ(n) is a power of two.

Folding reaches it exactly when φ(n) has no prime factor larger than three.

Both conditions are computable in a moment, which is why the figure above is a calculation rather than a list. The generator factors the totient of each n and reports what is left after the permitted primes are divided out; nothing is recalled from a table.

Run the test on seven. φ(7) = 6 = 2 × 3. Not a power of two, so no compass; no factor above three, so a fold. Run it on nine: φ(9) = 6, same answer. Run it on five: φ(5) = 4, a power of two, so both. Run it on seventeen: φ(17) = 16, which Gauss famously noticed at nineteen.

The nine-sided case is the one worth dwelling on, because it is the trisection in disguise. A nonagon is a triangle with each of its angles cut into three, and trisecting an angle by folding is exactly the operation that produces it. Anyone who has folded a trisection has folded two thirds of a nonagon without noticing; the same cubic underlies both, and the totient test simply records that fact in arithmetic.

There is a further pattern in the table that is easy to miss. Doubling n never changes the verdict, because φ(2n) is either φ(n) or twice it, and neither operation introduces a new odd prime. So the constructible polygons come in doubling families: three, six, twelve, twenty-four; five, ten, twenty; seven, fourteen, twenty-eight. Bisecting an angle is the one operation every tool has, which is why every row of the table is closed under it.

The cubic that gives the heptagon #

Naming the polynomial makes the whole thing concrete, and it is a pleasingly small one.

The three numbers cos 2π/7, cos 4π/7 and cos 6π/7 are the three roots of

8x³ + 4x² − 4x − 1 = 0.

That is a genuine cubic — it does not factor over the rationals, which is exactly why the compass fails — and its three roots are the three cosines together, not one of them at a time. Solving it produces all three, and any one of them locates the heptagon.

Folding solves it in one crease, and the mechanism is worth seeing rather than asserting. A fold that carries a point onto a line is tangent to the parabola with that point as focus and that line as directrix, because a parabola is exactly the set of points equidistant from a focus and a directrix and a fold is exactly a reflection. So a fold carrying two points onto two lines at once is a line tangent to two parabolas — and two parabolas have three common tangents.

Three tangents, three roots. The correspondence is not an analogy; the slopes of the tangents are the roots, and reading a cubic’s coefficients off as the two parabolas’ foci and directrices is a mechanical translation.

The translation is short enough to give. Tangents to the parabola with focus (u, a) and directrix y = −a are exactly the lines y = mx − mu − am²; tangents to the one with focus (b, v) and directrix x = −b are exactly y = mx + v + b/m. Setting the two intercepts equal and clearing the fraction gives am³ + um² + vm + b = 0. So the four coefficients of any cubic are the four numbers describing the two parabolas, read off in order, and the fold that solves it is the common tangent.

For the heptagon that means the two parabolas have foci at (½, 1) and (−⅛, −½), with directrices y = −1 and x = ⅛. Nothing about those numbers looks like a seven-sided figure. They are the coefficients of the cubic, divided through by eight.

Which theorem was checked, and how #

Two separate things are verified here, and neither is taken on trust.

The parabola construction is checked by tangency. The generator finds the roots of the stated cubic by bracketing and bisection, then computes, for each root, the intercept the tangent must have to touch the first parabola and the intercept it must have to touch the second. If those disagree by more than a part in a billion it throws. So the claim “this line touches both curves” is a measured claim rather than a drawn one.

The constructibility table is checked by construction, in the sense that it is computed rather than transcribed. Each cell is the result of factoring φ(n) and dividing out the permitted primes; a change to either rule changes the picture.

What neither figure can show is the proof that these tests are complete — that nothing outside the permitted degrees is reachable. That is Wantzel’s theorem for the compass and Pierpont’s for the fold, and both are arguments about field extensions rather than about pictures. A figure can show that seven is reachable by folding. It cannot show that eleven is not.

The polygons neither tool reaches #

The gap between the rows is the interesting part, and so is what lies beyond both.

φ(11) = 10 = 2 × 5. The five is fatal to both tests: not a power of two, and larger than three. The regular hendecagon is out of reach of straightedge, compass and single folds alike, and it is the first polygon of which that is true.

The same fate meets 23, 25, 29 and 31. There is nothing exotic about them; the totient simply picks up a prime factor of five or more, and both tools stop.

This is where the story usually ends, and it should not, because the restriction to one fold at a time is doing the work. Allowing two simultaneous folds solves quartics, three solves quintics, and the hierarchy continues — so the hendecagon is reachable by a two-fold construction, and every regular polygon whatever is reachable if enough creases are made in a single motion. The wall at eleven is a wall around a particular idealisation of what a folder does, not around folding.

Where the model stops #

Exactness is about the mathematics, not the paper. The construction is exact in the sense that the fold line lies where the cubic says. A folded heptagon is as accurate as the folding, which for a sheet of ordinary paper means a fraction of a degree at best, and the accumulated error over seven vertices is visible.

One fold at a time. The whole classification above assumes single folds. Lifting that assumption changes the answer completely, as above, and the assumption is exactly the one that makes the axiom list finite.

Nothing is said about the folding sequence. Knowing that cos 2π/7 is constructible by a fold does not by itself give a procedure a person can follow. The literature contains explicit heptagon constructions; deriving one is a separate exercise from proving one exists, and the gap between the two is the same gap that separates knowing a crease pattern folds flat from knowing how to fold it.

The circle is not drawn. A construction of the regular n-gon by folding produces the angle; realising the polygon as a shape on the sheet is a further step. The figures here mark the angles rather than the finished polygon, which is honest about what has been constructed.

Degrees, not fields. The tests above are stated in terms of the degree of a single cosine. The full statement is about the field generated by all of them, and for these particular numbers the two coincide — but the coincidence is a theorem rather than an obvious step.

Paper is not the plane. Every construction here assumes a sheet of unbounded extent, or at least one large enough for the fold lines to land on. A real square runs out, and a construction whose fold line leaves the paper is not a construction. The figures on this site clip fold lines to the sheet for exactly that reason, and it is one of the idealisations that cost something in practice.

The surprise: the same numbers turn up in prime theory #

The two lists have names that come from somewhere else entirely, and the connection is not decorative.

A prime p passes the compass test exactly when p − 1 is a power of two, which makes it a Fermat prime. Five are known — 3, 5, 17, 257 and 65537 — and no sixth has ever been found. Whether the list is finite is open.

A prime passes the folding test exactly when p − 1 has no factor above three, which makes it a Pierpont prime. These are much commoner: 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163 and onward, and it is conjectured, though not proved, that there are infinitely many.

So “which polygons can be folded” is a question whose answer depends on an unsolved problem in number theory. That is an unusual position for a subject about paper to be in, and it is the reason the heptagon story is worth more than a footnote: the tool determines a class of primes, the class of primes determines the polygons, and one of those classes may well be finite while the other is probably not.

Who found it, and when #

The dates are unusually far apart, which is part of why the result feels like two separate stories.

Gauss found the seventeen-gon in 1796 and stated the general condition in the Disquisitiones five years later, giving the sufficiency: an n-gon with the right totient can be built. He asserted the necessity as well and did not prove it. Pierre Wantzel supplied the proof in 1837, in the same paper that settled trisection and the doubling of the cube.

James Pierpont published the corresponding classification for the wider family of numbers in 1895, long before anybody connected it to paper. Margherita Piazzolla Beloch found the fold that solves cubics in 1936. Robert Lang and others assembled the pieces into the statement about origami constructions in the 1980s and 1990s.

So the arithmetic was finished a century before the geometry it describes was recognised, and the two halves of the heptagon’s story were written by people who never knew they were working on the same problem.

Beloch’s case is the one that rewards attention. She was a professional geometer working on ruled surfaces, and she came to paper through a puzzle book: Sundara Row’s Geometric Exercises in Paper Folding, published in Madras in 1893, which is the first serious treatment of folding as mathematics and which Beloch read in Klein’s German edition. Row had shown that folding does the ordinary Euclidean constructions. Beloch saw that it does more, found the fold that solves cubics, and published it in a journal that the origami community did not read for fifty years.

That is a recurring shape in this subject. The tree method had a mathematical statement long before it had a name; the Miura fold was published as engineering rather than as folding. Results arrive in one field and are noticed in another a generation later, and the delay is usually the time it takes for somebody to be fluent in both.

What a folded heptagon is actually worth #

A construction that exists and a construction that helps are different things, and it is worth asking which this is.

For making a seven-sided piece of paper, it is not much help. A protractor is faster, a printed template is faster still, and the accumulated error over seven creases is worse than either. Nobody folds a heptagon because they need a heptagon. Compare it with Haga’s construction, which divides a square into any whole number of parts and is genuinely the fastest way to do that with no tools — a result of the same kind that earns its keep in a way this one does not.

For understanding what folding is, it is the whole point. The heptagon is the smallest object that separates the two toolsets, so it is the cheapest possible demonstration that a fold is not a convenient compass. Every other elementary construction — bisecting, perpendiculars, dividing a square into equal parts — can be done both ways, and a reader who has only met those has no reason to believe the tools differ at all.

That is why the heptagon appears in every account of the subject and why it appears early. It is not the most useful thing folding does. It is the first thing folding does that nothing else can, and there is no smaller example.

The ladder from here #

Later rungs against this anchor: Wantzel’s proof, and what “degree is a power of two” is really saying about a tower of extensions. Gauss’s construction of the seventeen-gon, which is a computation rather than a drawing. An explicit folding sequence for the heptagon, and how much of it is axiom 6 and how much is bookkeeping. The hendecagon by a two-fold construction. Pierpont primes and what is known about their density. Constructions on a cone or a cylinder, where the reachable set changes because the reflections do. And the practical question of accuracy — which of the published heptagon constructions actually folds most accurately, which is an experiment rather than a theorem and does not appear to have been done.

Seven sides, one crease, and a two-hundred-year gap between the proof that it cannot be done and the observation that it can.