Kissing numbers, sphere packings and the number 24
2 segments around one, 6 circles around one, 12 spheres around one, 24 hyperspheres around one: the kissing numbers of the first four dimensions.
The kissing number of a dimension is the largest number of equal spheres that can touch another sphere at the same time without overlapping. More simply: how many neighbouring spheres fit at most around a central one?
- Dimension 1: the “sphere” is a segment. One more fits at each end. Kissing number 2.
- Dimension 2: exactly six equal circles fit around a circle. Kissing number 6.
- Dimension 3: exactly twelve fit around a sphere. Kissing number 12.
- Dimension 4: only at the beginning of the 21st century was it proved that the number is 24.
The kissing numbers are also known in 8 dimensions (240) and 24 dimensions (196,560). In 24-dimensional space the spheres sit on the Leech lattice, so densely that no room is left. For all other dimensions only bounds are known. There is no general formula.
In the plane: six circles around a seventh. This is a section of the densest circle packing.
Twelve around one
In space the twelve neighbouring spheres lie as in the face-centred cubic lattice. Joining their centres gives a cuboctahedron. It can be read as four interlocking hexagons, 4 × 6 = 24, and it has exactly 24 edges.
Twelve spheres around a thirteenth. Their centres form a cuboctahedron with 12 vertices and 24 edges.
Why 24 matters in space
The faces in the crystal lattices of sphere packings are exclusively triangles, squares or hexagons made of triangles. The most important solids of these lattices:
| Solid | Faces | Product |
|---|---|---|
| Cube | 6 squares | 6 × 4 = 24 |
| Octahedron | 8 triangles | 8 × 3 = 24 |
| Stella octangula | 24 triangles | 24 |
| Cuboctahedron | 8 triangles + 6 squares | 3 × 8 = 4 × 6 = 24 |
| Tetrahedron | 4 triangles | 4 × 3 = 12 = 24 : 2 |
Because the lattices are true fractals, three-dimensional Sierpiński tetrahedra and octahedra, in the first step every face divides into as many sub-faces as it has vertices. The octahedron thus gets 3 × 8 = 24 sub-faces, the cube 4 × 6 = 24, the tetrahedron 3 × 4 = 12. Cut the tips off the first-stage octahedral fractal and you get a cuboctahedron again.
If the geometry of gap-free space filling is tied to 24 in so many ways, the question arises: is this highly composite number connected not only with the distribution of the primes (see The number 24) but with space itself? In the ideas of Peter Plichta, too, 24 plays the leading role in connection with a four-dimensional space, although Plichta does not mention kissing numbers or sphere packings. His theories are more than 20 years older than the proof of the kissing number 24.
The kissing numbers in Christian symbolism
A striking series from Christian tradition stands alongside the kissing numbers:
| Dimension | Kissing number | Symbolism |
|---|---|---|
| 1 | 2 around 1 | the Trinity: Son and Spirit around the Father |
| 2 | 6 around 1 | six days of creation and the seventh as the day of rest |
| 3 | 12 around 1 | twelve disciples around Jesus as the thirteenth |
| 4 | 24 around 1 | the 24 elders around the throne of God in the Revelation of John |
Further readings: three creates the rhythm of six of the twin primes. Seven counts as a divine number, with seven days of the week and Sunday as the Lord’s day. Thirteen stands for transformation and the overcoming of death. 24 recalls Christmas Eve, 25 Christmas Day, on which, in this reading, the divine enters matter. 25 is no longer a prime, but 5 × 5. Primes, divisible only by themselves and one, become an image of unity here. The numbers 1 × 2 × 3 create a “divine order” with the rhythm of six. Five shatters this crystalline order again and creates variety. The pentagram stands for the golden ratio, regarded as the principle of life, and five as the number of the human being.
So, this time truly across four dimensions, a tetractys appears again: 1 + 2 + 3 + 4 = 10 and, as with the cuboctahedron, 6 + 6 + 6 + 6 = 24 = 1 × 2 × 3 × 4. This matches the four rhythms of six of the twin primes up to 24. Here, as with the 5-cell, the fourth dimension stands for entry into the transcendent and for the embodiment of spiritual principles. The Kabbalah says: Malkuth is Kether and Kether is Malkuth. The divine reveals itself in matter. Matter reveals the divine.
Continued under The throne of God and the number 24.