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 and synthesis, or father, mother and child. In symbolism it stands for energy and the masculine.
- The square is the sign of earth: resting, firm and 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.
Perfect symmetry
Is there anything more primal than the circle and the sphere? Every point source of energy, of light or heat for instance, radiates equally in all directions and so creates a spherical boundary around its centre of its own accord. An atom, too, forms a kind of sphere with its electron shell.
Symmetry is the property of a figure to map onto itself under certain movements, that is, to appear unchanged. Perfect symmetry is shown by the circle in the plane and the sphere in space: they have infinitely many axes of symmetry through their centre. Conversely, circle and sphere are most closely connected precisely with the regular figures that have the fewest axes of symmetry: triangle and square in the plane, tetrahedron, octahedron and cube in space. This page follows that relationship. For number theory one has to start from symmetric figures, regular polygons with equal sides and equal angles.
What follows looks trivial, you can see it in any tile showroom. Precisely for that reason it pays to look at it consciously. Basic principles of number theory, even and odd numbers or the twin primes in the rhythm of six, show up in these simple figures and then repeat without end. For this is about: number theory + geometry = philosophy.
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. Of the three, the hexagon needs the least boundary for the same area. That it is even the most economical of all conceivable divisions of the plane into equal areas was proven by Thomas Hales only in 1999. Whether bees build their combs of hexagons for this reason or whether the wax forms itself that way is disputed (see The honeycomb).
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 copies of a single other one, 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). All other polygons, the heptagon, nonagon, decagon, hendecagon and everything from the 13-gon on, do not occur in such tilings. Related to 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, together with two, produces the rhythm of six in the sieve of Eratosthenes, in which all primes are arranged. Every prime from five on stands right next to a multiple of six, and twin primes stand on either side of one. 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 apart from 24 itself. 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 polygons from 1 (point) to 12 in three groups of four. Highlighted are those that occur in regular and Archimedean tilings: all in the first group, every other one in the second, only the fourth in the third. Together they are the divisors of 24.
The same divisors shape space as well. Among the Platonic solids 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 composite number with no figure of the type “compound star” whose quotient exceeds 2 (primes have none anyway). 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 two of the three numbers whose polygons tile by themselves.
Triangle and square, the odd ones out
Triangle and square keep turning up on this site as exceptions: in tiling the plane, in the fractal polygons, where the square is the last fractal without overlap, in the simplexes and in number theory, down to the gaps between twin primes. They show with particular force how everything is connected with everything.
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. The six triangles in the hexagon have three full circles together, the hexagon itself two. 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.
In space, too, the crystal lattices of sphere packings can be built only from triangles and squares. The lattice of the densest packing, Kepler’s sphere packing, shows a tetrahedron from one side, a pyramid on a triangular base, and half an octahedron from another, a pyramid on a square base. Both belong to one and the same structure. It is also known as Buckminster Fuller’s isotropic vector matrix. Its gaps consist alternately of tetrahedra and octahedra, and it can be divided self-similarly into ever smaller tetrahedra and octahedra. Kepler was the first to publish on it, in 1611, after Thomas Harriot had already studied the question. Anyone who wants to understand the structure should best build a model of it.
The face-centred cubic lattice of the densest sphere packing. From one direction you see triangles, from another squares.
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.
Sources and further reading
Sources
- Johannes Kepler, Harmonices mundi libri V, Linz 1619, book II (tilings)
- Thomas C. Hales, The Honeycomb Conjecture, Discrete & Computational Geometry 25, 2001, pp. 1 to 22
- Heinrich Nollius, Theoria philosophiae hermeticae, Hanau 1617
Further reading
- Holger Ullmann, tetraktys.de, since 2010