Hexagon

The regular hexagon has the simplest construction of any polygon, and it requires no measuring whatever. Draw a circle. Without altering the compass setting, place the point anywhere on the circumference and strike an arc across it; move to that crossing and repeat. Six steps bring you exactly back to the start, and joining the six marks gives a regular hexagon.

This works because the side of a regular hexagon inscribed in a circle equals the radius. Each vertex, its neighbour and the centre form an equilateral triangle, and six of those fill the 360 degrees about the centre at 60 degrees each. Euclid gives the construction at Book IV Proposition 15 and notes the same fact.

The hexagon is constructible under Gauss-Wantzel because 6 is 2 times 3: a power of two multiplied by a single Fermat prime. Bisecting its arcs gives the 12-gon, the 24-gon and so on indefinitely. This is also why the hexagon and the equilateral triangle are so closely bound together. Joining alternate vertices of a hexagon gives a triangle, and superimposing both such triangles gives the six-pointed star, whose interior is again a hexagon.

Its interior angle is 120 degrees, the largest of any polygon that tiles the plane alone, and that is the reason three hexagons meet at every vertex of a honeycomb rather than four or six. Among all polygons that tile the plane, the hexagon has the fewest neighbours meeting at each corner, which turns out to matter a great deal for efficiency.

Only three regular polygons tile the plane by themselves: the equilateral triangle, the square and the hexagon. The proof is immediate. Copies meeting at a point must have interior angles summing to 360 degrees, so the interior angle has to divide 360 exactly. The interior angles of regular polygons run 60, 90, 108, 120, 128.57 and upward, and from 120 onward nothing divides 360 again. Only 60, 90 and 120 work.

Among those three the hexagon is the efficient one, and the claim that it is has both a name and a date. The honeycomb conjecture states that a regular hexagonal grid is the way to divide a surface into regions of equal area using the least total perimeter. Varro states something close to it in 36 BC and it is often credited to Pappus in the fourth century AD, which makes it one of the oldest conjectures in mathematics. Laszlo Fejes Toth proved it in 1943 for the restricted case in which every cell is a convex polygon. The general case, allowing cells with curved boundaries of any shape at all, resisted until Thomas Hales proved it in 1999. The key step was showing that the advantage a cell gains by bulging outward never outweighs what its neighbour loses by bulging inward.

Two cautions belong here. First, the theorem concerns dividing a plane, not honeycombs; the three-dimensional question of the most efficient way to close the end of a honeycomb cell is a different problem, and the arrangement bees actually use is very slightly less efficient than the best known. Second, real honeycomb hexagons are partly the result of wax flowing and settling under the bees' body heat rather than being built hexagonal from the outset, and how much of the pattern is behaviour and how much is physics is still investigated.

Kelvin's related question, the best way to partition three-dimensional space into equal-volume cells, remains open. Kelvin's own answer of 1887 stood unbeaten until Weaire and Phelan found a better one in 1994, and nobody has proved theirs optimal either.

Snowflakes are six-sided, and Johannes Kepler wrote the first serious attempt at explaining why: a short book of 1611, Strena seu de nive sexangula, offered as a new year's gift, in which he asked what makes every flake six-cornered and proposed that it arose from the packing of small identical units. He had no way to confirm it. He was right in outline, since the hexagonal symmetry does come from the arrangement of water molecules in ice, and the same short book contains the sphere-packing conjecture that carried his name until Hales proved that as well.

Hexagonal structures recur because they are what equal units under pressure tend to produce. Basalt columns at the Giant's Causeway and at Fingal's Cave crack into rough hexagons as lava cools and contracts. Insect compound eyes, some bird bone, graphene and the cells of a beehive share the pattern for related reasons of packing and economy.

In Judaism the six-pointed star, the Magen David, is two overlapping equilateral triangles with a hexagon at the centre. Its adoption as the principal Jewish symbol is late. The shape appears in many contexts, Islamic and Christian among them, for centuries without any specific Jewish meaning, its use as a communal emblem is documented in Prague from the fourteenth century, and it spread widely only in the nineteenth. Anyone dating it to Solomon or David is repeating a legend.

In Islamic geometric ornament the hexagonal grid is one of the principal underlying frameworks, generated in practice by exactly the compass method described above and elaborated into star patterns of six, twelve and twenty-four points.

Chinese divination uses the sixty-four hexagrams of the Yijing, figures of six lines each, though these are stacked lines rather than hexagonal geometry, and the connection to the polygon is verbal rather than formal.

Sources Strena Seu de Nive Sexangula (On the Six-Cornered Snowflake)Johannes Kepler with The Star of David: History of a Symbol, in The Messianic Idea in JudaismGershom Scholem, The Topkapi Scroll: Geometry and Ornament in Islamic ArchitectureGulru Necipoglu, Fathoming the Cosmos and Ordering the World: The Yijing (I Ching, or Classic of Changes) and Its Evolution in ChinaRichard J. Smith