Remarkable facts about the number 24
1 × 2 × 3 × 4 = 24. Four rhythms of six before 25, 24 symmetries of the tetrahedron, 24 edges in the primal fractal and the kissing number of the fourth dimension.
The tetractys has two faces, the sum and the product:
1 + 2 + 3 + 4 = 10
1 × 2 × 3 × 4 = 24
24 is the last number before 25 = 5 × 5, the point at which the order of the twin primes breaks open. 24 turns up in an astonishing number of places.
In number theory
- Four rhythms of six. Up to 24 there are exactly four rhythms of six, and in all of them the twin positions are fully occupied by primes. The disorder begins with 25.
- Seven primes. Up to 24 exactly seven primes lie in the rhythm of six (5, 7, 11, 13, 17, 19, 23). With 1, 2 and 3 that makes ten.
- p² − 1. Take a prime greater than 3, square it and subtract 1, and the result is always divisible by 24: 5² − 1 = 24, 7² − 1 = 48, 11² − 1 = 120. This follows from the rhythm of six, since every such prime is 6n ± 1.
- Prime number cross. In rings of 24, all primes from 5 on lie on eight rays (see Prime number cross).
The first eight rhythms of six. Up to 24 every twin position is a prime. 25 is the first empty place.
In geometry
- The primal fractal of the tetrahedron has 10 vertices and 24 edges (see Gap-free filling of space).
- The cuboctahedron, the arrangement of twelve spheres around one, has 24 edges. The stella octangula has 24 triangular faces.
- Cube and octahedron: 6 × 4 = 8 × 3 = 24.
- Tilings: all polygons that occur in regular and Archimedean tilings have a number of vertices that divides 24: 3, 4, 6, 8, 12 (see Circle, triangle and square).
- Kissing number: in four-dimensional space exactly 24 spheres touch another one (see Kissing numbers).
Symmetries of the tetrahedron and group theory
Group theory, the foundation of modern research into symmetry, offers another approach. A “group” there is the set of all rotations and reflections that map a figure onto itself, leaving it unchanged. Complex symmetries can be composed of several groups.
The tetrahedron has 12 rotations. Together with the 12 transformations that involve reflections (6 plane reflections and 6 rotoreflections), that makes 24. Counted another way: the triangle has 3 rotations and 3 reflections, and times the 4 faces of the tetrahedron that again gives 24. These are exactly all the ways of arranging the four vertices: 1 × 2 × 3 × 4.
Related topics and interfaces: the “anatomy of space” with the cubic crystal system, the Leech lattice in dimension 24, and the order of the primes. There is also a connection in thought with the Sefer Yetzirah, in which permutation and combinatorics, the tools of group theory, are at the centre of an act of creation. Two theses follow from this: that the Jewish secret teaching is very close to Platonism and Pythagoreanism, and that ancient thinkers may already have applied “modern” group theory consciously.
In symbolism and the calendar
The 24 hours of the day, the 24 elders of Revelation, 24 December and 24 June as St John’s Day: these traces are followed up in The throne of God and the number 24. A sober objection must be kept in mind: whoever invents a division of time will, for practical reasons, choose an even, highly composite number to avoid fractions. 24 would be a favourite without any deeper reasons.