Heptagon

The regular heptagon is the first regular polygon that compass and straightedge cannot construct. Not the first that is difficult, the first that is impossible, in the strict sense that no finite sequence of the permitted operations can produce it.

The governing result is the Gauss-Wantzel theorem: a regular polygon with n sides is constructible with compass and straightedge if and only if n is a power of 2 multiplied by a product of distinct Fermat primes. A Fermat prime has the form 2 to the power 2 to the power k, plus one, and only five are known: 3, 5, 17, 257 and 65537. Seven is prime, but it is not a Fermat prime and it is not a power of two. The heptagon fails the condition.

Gauss supplied the sufficient half of the theorem and published it in the Disquisitiones Arithmeticae of 1801, having recorded on 30 March 1796, a month before his nineteenth birthday, that the regular 17-gon is constructible. That was the first advance on the classical list of constructible polygons in over two thousand years, and he is said to have wanted a 17-gon on his gravestone; the monument at Brunswick carries a seventeen-pointed star instead, the mason having observed that a 17-gon would be mistaken for a circle. Gauss asserted the necessary half without proving it. Pierre Wantzel supplied that proof in 1837, and it is Wantzel's contribution that converts the statement from a criterion for success into a demonstration of impossibility.

The algebra behind it is concrete. Every point reachable by compass and straightedge comes from intersecting lines and circles, and each such step solves at worst a quadratic, so every constructible length has degree a power of 2 over the rationals. Constructing a regular heptagon requires the quantity 2 cos of 2 pi over 7, whose minimal polynomial is the cubic x cubed plus x squared minus 2x minus 1, irreducible over the rationals. Degree 3 is not a power of 2. There is nothing further to be said: the number lies out of reach of the tools.

Impossible does not mean undrawable. It means undrawable exactly with those two instruments, and the workarounds are old and good.

The best-known approximation is Albrecht Durer's, published in his Underweysung der Messung of 1525. Inscribe an equilateral triangle in a circle; half the length of the triangle's side, stepped around the circumference, is very nearly the side of the inscribed heptagon. The error is a fraction of one per cent, far below anything a mason or an engraver could detect, and Durer presents it as a working method rather than as a theorem. His book gives an approximate nonagon in the same spirit.

Exact constructions become available once the restriction is loosened. Archimedes produced a heptagon by neusis, sliding a marked ruler until two points on it meet two given curves at a specified separation; the construction survives through Arabic transmission and is regarded as one of the finest of antiquity. Neusis is strictly more powerful than compass and straightedge, since it solves cubic equations, and it therefore also trisects the angle and doubles the cube. Paper folding does the same work: the origami axioms permit cubic solutions, and an exact heptagon can be folded. A conic section drawn as a curve, or the quadratrix, will also deliver one.

None of this contradicts the theorem. Each method adds an operation that the classical rules exclude. What the theorem states is precisely how much power those two instruments have, and the heptagon marks the boundary.

Seven sides therefore appear rarely in traditional architecture, where the compass governed the drawing board. The most familiar seven-sided object in ordinary use is a curved one. The British fifty pence coin is an equilateral curve heptagon, a shape of constant width, so it rolls smoothly through slot machinery despite having no circular edge; the seven-sided form was chosen so that it could not be confused by touch with any other coin.

Seven is heavily loaded in religious thought, and it is worth noting that its weight came from the sky rather than from geometry.

The classical world counted seven moving bodies against the fixed stars: Sun, Moon, Mercury, Venus, Mars, Jupiter and Saturn. That count fixed the seven-day week, the naming of the days, the seven metals of alchemy, and much of the planetary correspondence that later esoteric systems inherited. Babylonian astronomy is the source, and the transmission through Greek and Roman practice into the calendar is well documented.

In the Hebrew Bible seven governs the structure of the creation account, the seventh-day sabbath, the sabbatical year, the jubilee after seven sevens, and the seven-branched menorah of the Temple. Revelation is built on sevens throughout: churches, seals, trumpets and bowls. In Islam there are seven heavens, seven circuits of the Kaaba, and seven verses in the opening sura.

Indian tradition counts seven chakras in the systems now most widely known in the West, though that is not the only enumeration in the source texts, several of which give four, five, six or a different number, and the standard modern seven-chakra scheme owes a good deal to how the material was received and simplified in the twentieth century. Seven Hathors appear in Egyptian texts as determiners of fate, and the seven sages, the saptarishi, name the stars of the Plough in Indian astronomy.

The heptagram or seven-pointed star is used in several modern traditions, as an emblem in Thelema, in some Wiccan practice, and in modern devotional design. It has no significant premodern architectural presence, for the straightforward reason that nobody could draw one exactly.