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.
Why p² − 1 is divisible by 24
The proof is short and elegant. By the difference of squares,
p² − 1 = (p − 1) · (p + 1)
Since p > 3 is prime, p is odd, so p − 1 and p + 1 are two neighbouring even numbers. Of two neighbouring even numbers one is divisible by 4, so the product is divisible by 8. Moreover, of the three consecutive numbers p − 1, p and p + 1 one is divisible by 3, and it cannot be p. So the product is also divisible by 3, altogether by 24. Since 5² − 1 = 24, there is no larger divisor that always fits. (After a presentation by Werner Brefeld, see Links.)
The quotients (p² − 1) : 24 are 1, 2, 5, 7, 12, 15 … These are the generalised pentagonal numbers n · (3n − 1) : 2 for n = 1, −1, 2, −2, 3, −3 … Writing p = 6n ± 1, this follows directly. Whether these pentagonal numbers have anything to do with the pentagram, which in the simplexes opens the disorder right after 24 (2.5 · 10 = 25), would be worth investigating. 5 is the first prime on a twin position 6n ± 1, 25 the first non-prime on such a position.
Squares, rings and 24
Nest squares of odd side length inside one another, and each ring around the square of side 2k − 1 has exactly 8k cells: 8, 16, 24, 32 … Now leave out all side lengths divisible by three, and exactly the twin positions 1, 5, 7, 11, 13, 17, 19 … remain, including 1 but without 2 and 3. The cells between two consecutive such squares are always a multiple of 24:
| Sides | cells between | divided by 24 |
|---|---|---|
| 1 → 5 | 24 | 1 |
| 5 → 7 | 24 | 1 |
| 7 → 11 | 72 | 3 |
| 11 → 13 | 48 | 2 |
| 13 → 17 | 120 | 5 |
| 17 → 19 | 72 | 3 |
| 19 → 23 | 168 | 7 |
| 23 → 25 | 96 | 4 |
Alternately they give the sequences 1, 2, 3, 4 … (summed, the triangular numbers) and 1, 3, 5, 7 … (summed, the square numbers). Two basic forms of figurate number are thus contained in a single figure.
Pyramidal numbers and 24
Stack spheres into a square pyramid, like cannonballs or oranges, and you get the pyramidal numbers 1, 5, 14, 30 … with the formula
P(n) = n (n + 1) (2n + 1) : 6
For n = 24, P(24) = 24 · 25 · 49 : 6 = 4900 = 70², because 24 : 6 = 4 and 25 and 49 are themselves squares. A stack of 4900 balls can therefore also be laid flat as a square of 70 × 70. Édouard Lucas asked in 1875 whether there are other such cases, and G. N. Watson proved in 1918: apart from 1, 24 is the only height at which the pyramid gives a square number.
Fibonacci numbers and 24
Take the single-digit digit sums of the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21 …, and the sequence repeats without end in a cycle of 24:
Set the first twelve against the second twelve and each pair adds up to 9, except at the end, where two nines stand one above the other. Digit sums depend on the counting system, though: this is casting out nines, since in the decimal system a number and its digit sum always differ by a multiple of 9. Because this site deals precisely with the “decimal coding” of the simplexes and its link to 24, the fact still deserves attention. The Fibonacci sequence in turn appears in Pascal’s triangle and is linked with the golden ratio (see Pentagram and golden ratio).
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.
It is striking that Peter Plichta also links 24 with a four-dimensional space and gives the Sierpinski triangle an important role, yet does not mention kissing numbers and sphere packings at all. His theories are older than the proof that the kissing number in four dimensions is exactly 24 (Oleg Musin, 2003). Whether there is a direct connection here would be worth investigating. Finally, the two faces of the tetractys side by side once more: 1 + 2 + 3 + 4 = 10 and 1 × 2 × 3 × 4 = 24.
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.