Six in the shape itself
Six has a geometric quality that few other numbers share, and it is a matter of mathematics rather than belief. A regular hexagon can be cut into six equilateral triangles that meet at the centre. As a result, each side of the hexagon is exactly as long as the distance from the centre to a corner. Take a compass, keep it fixed at that radius, and step around a circle: after six steps you return to the start. This is one of the oldest and simplest constructions in geometry, and it is the basis of many ornamental patterns that look like flowers.
The same fact gives the hexagon some practical uses. The interior angle of a regular hexagon is 120 degrees, and three of them meeting at a point make 360 degrees, so hexagons fit together with no gaps. Only three regular polygons can tile a flat surface on their own: the triangle, the square and the hexagon. Among these, the hexagon encloses the most area for a given edge length.
The hexagon also played a modest part in the history of measurement. A hexagon drawn inside a circle has a perimeter of six radii, which is shorter than the circumference of 2π radii, so π must be greater than 3. Archimedes began with this figure, doubled the number of sides several times up to a 96-sided polygon, and so produced one of the best-known approximations of pi from antiquity. The division of the circle into 360 degrees, a number divisible by six, is usually linked to the Mesopotamian sexagesimal system, though exactly how the two connect is debated.
Circles, packing and the honeycomb
Place six identical coins around a seventh, each touching the centre one and its two neighbours, and they fit exactly. In the plane, six is the largest number of equal circles that can touch a central one. Repeating the pattern produces the hexagonal arrangement, and it is the densest possible way to pack equal circles in a plane, filling about 90.7 per cent of the area. That is a classical theorem of geometry, with the proof credited to Axel Thue and László Fejes Tóth.
The honeycomb is the well-known natural example. Honeybee cells are hexagonal prisms, and the pattern uses a minimum of wax for the storage space it provides. The ancient mathematician Pappus of Alexandria, writing in the fourth century CE, praised the bees for choosing the hexagon, which he argued encloses more than the square or the triangle for the same wall length. The general statement, that a hexagonal grid divides a plane into equal cells with the least total perimeter, is known as the honeycomb conjecture, and Thomas Hales proved it in 1999.
It is easy to overstate the bees’ role. Studies of how comb forms suggest that the hexagons may arise partly from the wax softening and the cells pushing against each other, rather than from the insects measuring angles. The mathematics is real; the idea of a bee as a geometer is a figure of speech.
Six-fold shapes also appear in snow crystals, which grow in six-armed patterns because of the way water molecules arrange themselves when they freeze. Johannes Kepler wrote a small essay in 1611 asking why snowflakes have six corners, and he guessed correctly that the answer lay in a regular packing of tiny parts, even though the atomic theory needed to confirm it was far in the future.
The six-pointed star
The hexagram is formed by two overlapping equilateral triangles, one pointing up and one pointing down. It is one of the most widely travelled symbols, and its meanings are separate in different places.
- Decoration. Six-pointed stars appear in the ornament of the ancient Mediterranean and Near East, and they were common in medieval Islamic art, where the star is built into tiling and woodwork as part of geometric design. Such uses were often decorative, with no single established meaning.
- Jewish tradition. The shape became linked with Jewish communities over time. It is often cited as appearing on a flag used by the Jewish community of Prague in the fourteenth century, and it spread more widely during the nineteenth century, when it was taken up as a communal emblem. It appears on the flag of Israel, adopted in 1948. As the Star of David, or Magen David, it is a symbol of identity more than a symbol of a number, and it was not a prominent religious symbol in early Jewish sources.
- Seal of Solomon. In Jewish, Christian and Islamic folklore, a seal ring attributed to King Solomon was said to command spirits. It was depicted at times as a hexagram and at other times as a pentagram, depending on the author and period.
- South Asia. In Hindu and Tantric imagery, the interlocked triangles are read as union of masculine and feminine principles or of two cosmic forces. The Sri Yantra uses a related but more complex design of nine triangles, and the shatkona, a six-pointed figure, is explained along these lines in some traditions.
- Alchemy and the West. Combining an upward triangle and a downward one was a way of expressing a union of opposites, such as fire and water.
All of these are separate uses that happen to share a shape, and it would be a mistake to read one tradition’s meaning into another’s.
Six in other geometric traditions
Pythagorean thinkers had their own reasons for valuing six, mainly its arithmetic property as a perfect number six, which belongs to a different story. The I Ching, the Chinese book of divination, uses hexagrams in a different sense: figures of six stacked lines, each broken or unbroken, giving 64 combinations (2 to the sixth power). The link to the later binary notation was noted in the early eighteenth century by Leibniz, who saw a parallel when a correspondent sent him an arrangement of the hexagrams. The six-line “hexagram” shares only a name with the six-pointed star.
What can be said fairly
Six-fold geometry is built into the way equal circles and equal cells fit together, which is why it feels natural and why it recurs in nature and ornament. The symbolic meanings, from union of opposites to communal identity, were added by people at particular times and places. Keeping those two layers apart explains both why the hexagon appears so often and why it has meant so many different things.