Pentagon

The regular pentagon is constructible, and the reason is that 5 is a Fermat prime: it is 2 to the power 2 to the power 1, plus one. Under the Gauss-Wantzel condition, which requires a side count equal to a power of 2 times a product of distinct Fermat primes, five qualifies directly. So do 10, 20 and 40 by repeated bisection, and 15 by combining the pentagon with the triangle, which Euclid does at Book IV Proposition 16.

Euclid's own route is indirect. He must first cut a straight line in extreme and mean ratio, so that the whole is to the greater part as the greater part is to the lesser. That is Proposition 11 of Book II, restated at Book VI Proposition 30, and it is the construction of what is now called the golden ratio, phi, equal to one plus sqrt 5 all divided by two, about 1.618. From it he builds an isosceles triangle whose base angles are each double its apex angle, the 72-72-36 triangle, inscribes that in a circle, and bisects to reach the pentagon at Book IV Proposition 11.

Shorter methods exist. The construction usually attributed to H. W. Richmond, published in 1893, inscribes a pentagon in a given circle in a handful of steps: take two perpendicular radii, bisect one of them, use that midpoint to bisect an angle, drop a perpendicular, and the resulting chord is the pentagon's side.

The ratio is not decorative. In a regular pentagon the diagonal divided by the side is exactly phi. Draw all five diagonals and they cut one another in the golden ratio, producing a pentagram whose interior is a smaller inverted pentagon, within which the same thing happens again, without end. That endless nesting is itself a proof that phi is irrational: if it were a ratio of whole numbers the regress would have to terminate, and it does not.

The five-pointed star drawn in one unbroken line is the pentagon's set of diagonals and nothing else. Its history is long and better documented than most sacred-geometry claims.

The Pythagoreans used it as a sign of recognition. Lucian, in the second century AD, describes the figure as the triple interwoven triangle used by the school and reports that they called it health, and a scholiast on Aristophanes says the same. The Greek word hygieia has five letters, and the practice of writing one at each point is attested on later amulets, with the Latin salus doing the same job. That the Pythagoreans greeted one another with a wish for health is independently reported. The claim that the pentagram served as their badge is therefore standard and reasonably well sourced, though it rests on writers living several centuries after the school itself.

The symbol is considerably older than the Pythagoreans. Pentagram signs appear in Mesopotamian administrative texts of the fourth and third millennia BC, apparently as a sign for something like a heavenly body or a district, and the original meaning is not securely recovered.

In medieval Christian use the pentagram is a protective sign and occasionally stands for the five wounds of Christ. The fourteenth-century poem Sir Gawain and the Green Knight devotes a long passage to the pentangle on Gawain's shield, calling it Solomon's sign and reading its five points as five sets of five virtues. It is one of the fullest explanations of a symbol that anyone in the period troubled to write down.

The inversion, the idea that a point-down pentagram signifies evil, is recent and traceable. It belongs to nineteenth-century French occultism, principally to Eliphas Levi, and reached popular culture through twentieth-century channels. It is not a medieval distinction, and reading it back into earlier material is simply an error.

A regular pentagon cannot tile the plane. The reason is arithmetic: its interior angle is 108 degrees, and 108 does not divide 360, so pentagons meeting at a point leave a gap of 36 degrees. Only three regular polygons tile alone, triangle, square and hexagon, because only 60, 90 and 120 divide 360 exactly.

Irregular convex pentagons are a different and much harder question, and it stayed open for a century. Karl Reinhardt found five tiling types in 1918. More appeared in the 1960s and 1970s, several of them found by Marjorie Rice, an amateur working from a Scientific American column, who discovered four new types. A fifteenth was found by computer search in 2015 by Casey Mann, Jennifer McLoud-Mann and David Von Derau. In 2017 Michael Rao announced an exhaustive computer search showing there are no others, and part of the proof was independently verified. The current position is that there are fifteen types, resting on a computation rather than on a short argument.

Fivefold symmetry is also the signature of a structure long held to be impossible in crystals. The crystallographic restriction permits two-, three-, four- and sixfold rotational symmetry and no other, because those are the only ones compatible with a repeating lattice. In 1982 Dan Shechtman observed a tenfold diffraction pattern in an aluminium-manganese alloy. He was disbelieved, and Linus Pauling opposed the result publicly for years. Quasicrystals, ordered but non-periodic and showing fivefold symmetry, are now an established class of material, and Shechtman received the Nobel Prize in Chemistry in 2011.

Fivefold form is common in living things: starfish and sea urchins, the flowers of the rose family, an apple cut across the core. It is rare in minerals. The old association of five with life rather than with stone is, for once, roughly borne out by where the symmetry actually occurs.