Nonagon

The regular nonagon cannot be constructed with compass and straightedge. The reason is a detail in the governing theorem that is easy to overlook and that does all the work here.

The Gauss-Wantzel theorem states that a regular n-gon is constructible if and only if n is a power of 2 multiplied by a product of distinct Fermat primes. Fermat primes are those of the form 2 to the power 2 to the power k, plus one; only 3, 5, 17, 257 and 65537 are known. Nine is 3 times 3. Three is a Fermat prime, which is why the equilateral triangle is constructible, but nine uses it twice, and the theorem does not permit repetition. That single word, distinct, is why the nonagon fails while the hexagon, 2 times 3, and the 15-gon, 3 times 5, both succeed.

The underlying reason for the restriction is algebraic. Each new point in a classical construction comes from intersecting lines and circles, which solves at worst a quadratic, so every constructible length has degree a power of 2 over the rationals. A repeated prime factor forces an irreducible cubic into the field extension, and 3 is not a power of 2.

This is a proof, not a gap in knowledge. Gauss gave the sufficient direction in 1796 and published it in 1801; Wantzel proved the necessary direction in 1837. The nonagon has been known to be impossible for nearly two centuries, and no method using only those two instruments will ever produce one. Every nine-sided figure drawn with compass and straightedge, however carefully, is an approximation to something the tools cannot reach.

The nonagon's impossibility has a familiar face. A regular nonagon has a central angle of 40 degrees. An equilateral triangle, easily constructed, has a central angle of 120. To get from the one to the other is to divide 120 degrees into three, which is to trisect an angle.

Trisecting an arbitrary angle is one of the three classical problems of Greek geometry, alongside doubling the cube and squaring the circle. It is impossible with compass and straightedge, and it was Wantzel's paper of 1837 that proved it, the same paper that settled the constructible polygons. The nonagon and the trisection are two statements of a single fact.

Note the qualification: an arbitrary angle. Plenty of particular angles trisect perfectly well. A right angle divides into three 30 degree parts without difficulty, because 30 degrees is constructible. What fails is a general method, and 120 degrees is a concrete instance of that failure.

Approximations are entirely serviceable in practice. Durer's Underweysung der Messung of 1525 gives one, using three lens-shaped vesica figures set on the vertices of an inscribed equilateral triangle together with a division of a radius into thirds. Note the distinction that makes: dividing a straight segment into three equal parts is elementary, and Euclid does it; it is dividing an angle into three that cannot be done. Durer's nonagon is accurate to well within the tolerance of any craft that would have used it.

Exact nonagons become available once the rules are relaxed. A marked ruler used in neusis will trisect an angle, as Archimedes knew, and so will paper folding, and so will several classical curves invented for the purpose, among them the conchoid of Nicomedes and the quadratrix. The nonagon is not beyond geometry. It is beyond two particular instruments, and the theorem states exactly why.

Nine is three times three, and much of its symbolic weight comes from being the intensification of a number that already carried weight.

The Bahai faith is the clearest modern case of a religion committed to the figure. Its Houses of Worship are required to be nine-sided, with a central dome and nine entrances, and only a handful exist worldwide, at Wilmette, Sydney, Kampala, Panama City, Frankfurt, New Delhi, Apia and Santiago. Every one of them is a nonagon in plan. The reason is numerical: in the abjad system of letter values the word Baha sums to nine, and nine is the highest single digit, read as completeness and unity. The nine-pointed star is used as an emblem for the same reason, though Bahai sources are careful to say that it is a convention rather than a revealed symbol.

Egyptian theology grouped the Ennead of Heliopolis, nine gods running from Atum through to Osiris, Isis, Set and Nephthys; the Egyptian term simply means the nine. Norse cosmology counts nine worlds joined by Yggdrasil, and Odin hangs on the tree for nine nights. Chinese tradition treats nine as the greatest yang number, hence nine dragons, the nine rows of nine door studs on imperial gates, and ninefold divisions of imperial ritual space. Indian astrology counts the Navagraha, nine planetary influences. Dante's Inferno has nine circles and his Paradiso nine heavens.

Whether any of these traditions was aware that the nine-sided figure resists exact construction is doubtful, and there is no evidence that any of them was. The nonagon's mathematical stubbornness and its symbolic reputation for completion are independent facts that happen to sit side by side.