Circle, triangle and square
Humanity's three oldest symbols are also the basis of every gap-free tiling, and the divisors of 24 decide which polygons can occur in it.
Circle, triangle and square are among the oldest symbols of all.
- The circle returns into itself. It stands for unity, perfection and infinity. With a point at its centre, for creation. In Zen Buddhism it is the sign of enlightenment.
- The triangle counts as the most stable form and as the image of threeness: thesis, antithesis, synthesis, or father, mother, child. In symbolism it stands for energy and the masculine.
- The square is the sign of earth: resting, firm, material, in symbolism the feminine.
Set the natural numbers side by side as polygons, and the first ones are the most distinctive. From about ten vertices on, the corners blur and the polygon approaches the circle. All the more striking that the triangle and the square, furthest from the circle, are the ones most directly connected with it.
The three regular tilings
Only three regular polygons fill the plane without gaps when laid edge to edge: triangle, square and hexagon. The reason: at every vertex the angles must add up to exactly one full circle.
Top: the three regular tilings, the red dot marks a vertex. Bottom: the only two ways to lay equal circles so that they touch regularly. Their centres form squares or triangles.
The hexagon is a special case: it is the only regular polygon made up of another, of six triangles. Triangles in turn can only tile when they alternate upright and inverted.
The pentagon lies exactly between two tiling polygons and is the first that cannot tile with itself. The octagon can at least be laid together with squares, the dodecagon with triangles or with squares and hexagons (semi-regular or Archimedean tilings). From the 13-gon on, nothing works. In crystallography this is the crystallographic restriction: periodic patterns can only contain rotations by 360°, 180°, 120°, 90° and 60°, symmetries of order 1, 2, 3, 4 and 6. Fivefold symmetry exists only in the “almost periodic” quasicrystals.
Parallels in number theory
These findings have number-theoretical counterparts:
- Three and six (tile by themselves): three produces the rhythm of six in the sieve of Eratosthenes, in which all primes are arranged. Primes stand as twins around the multiples of six. Just as triangles tile only by alternating upright and inverted, even and odd numbers alternate.
- Four (tiles by itself): four stands for the fourfold, fractal order called the tetractys here. The square is the only polygon whose angles make one full circle.
- Five and seven (do not tile): five is the first prime in the rhythm of six, seven its twin. With 25 = 5 × 5, five breaks open the order of the twin primes. With it begins disorder, and at the same time the variety of possible combinations. Think of Penrose tilings and quasicrystals.
The divisors of 24
The polygons that can occur in regular and Archimedean tilings have 3, 4, 6, 8 or 12 vertices. Together with 1 and 2 (point and line) these are exactly all the divisors of 24. 12 is divisible by 1, 2, 3, 4 and 6, matching all possible rotations of periodic patterns.
Once more an order of fours appears:
- In the first group of four, 1, 2, 3, 4, every polygon tiles.
- In the second, 5, 6, 7, 8, only every other one: 6 and 8.
- In the third, 9, 10, 11, 12, only the fourth: 12.
The same divisors shape space as well. Only tetrahedra and octahedra together, or cubes alone, fill it without gaps, and their faces, vertices and edges number 4, 6, 8 and 12. More under Gap-free filling of space.
One more observation on the dodecagon: 12 is the last number with no figure of the type “compound star” whose quotient exceeds 2. The decagon is the first number with such a figure (10 : 4 = 2.5). If all quotients are included, this constellation first appears at 6 : 4, for exactly the two numbers whose polygons tile by themselves.
The hexagon and the seventh circle
Exactly six equal circles fit around one circle. Seven in all. The centres of the six outer ones form a hexagon of six triangles. Two triangles, each with an angle sum of half a circle, complete each other to a full circle, the seventh at the centre. All the angles in the hexagon add up to three full circles. Read the net as a pattern of hexagrams, and each hexagram, formed of two triangles, has an angle sum of exactly one full circle. The square reaches the same full circle with its four right angles alone.
A model of creation
At the points where circles touch, a fractal chain can be set going: the edges of the triangles and squares are halved, the halving points become the centres of a new grid of circles of half the diameter, and so on. This works with triangles and with squares, in space also with the face-centred cubic lattice, but no longer from the pentagon on.
Circles in a hexagonal net: six around one, and the next ring around them. In 'sacred geometry' the pattern is known as the flower of life.