# Contents

## Idea

24 is the natural number that follows 23 and precedes 25. It occurs in modern systems of measurement for historical reasons, thanks ultimately to its having many prime factors

$24 \,=\, 2 \cdot 2 \cdot 2 \cdot 3$

and it arises in many circumstances in mathematics and physics, the relations between which are not always immediately apparent.

## Examples

### Modular arithmetic

Fix a natural number $n$, and consider those numbers $x$ whose squares are congruent to 1 modulo $n$:

$x^2 \equiv 1 (\mod n) \, .$

A necessary condition for satisfying this is for $x$ and $n$ to be coprime. For some choices of $n$, this condition is sufficient, i.e., every $x$ coprime to $n$ satisfies this congruence. In fact, this congruence is equivalent to $gcd(x,n) = 1$ if and only if $n$ divides 24.

### Cannonball problem

The cannonball problem asks what size square pyramids can be made by stacking spheres such that the total number of spheres is a square number. Apart from the trivial solutions involving 0 spheres or 1 sphere, the only solution is a pyramid 24 layers tall:

$0^2 + 1^2 + 2^2 + \cdots + 24^2 = 70^2 \, .$

### Leech lattice

The Leech lattice is a “universally optimal” packing of hyperspheres in $\mathbb{R}^{24}$ originally discovered in coding theory. Surprisingly, it is closely related to the cannonball problem, since it can be constructed from a Lorentzian lattice in two higher dimensions by a method that uses the fact that

$v = (0, 1, 2, \ldots, 24, 70)$

has zero norm under that metric.

### Mathieu groups

The largest Mathieu group, $M_{24}$, is the automorphism group of a Steiner system containing 24 blocks. $M_{24}$ is also the automorphism group of the binary Golay code, an abelian group which is a 12-dimensional subspace of the vector space $\mathbb{F}_2^{24}$.

The binary Golay code can be used to construct the Leech lattice. Essentially, each codeword defines a point in 24-dimensional space, and we can fill in and build out this set. In more detail: Let $c$ be a Golay codeword, scale it by a factor 2, and add either $4x$, where $x$ is a vector in $\mathbb{Z}^{24}$ whose components sum to an even number, or $1 + 4y$ where $y \in \mathbb{Z}^{24}$ and its components sum to an odd number.

### 24-cell and the binary tetrahedral group

The 24-cell is a four-dimensional regular polytope with 24 vertices. Interpreting these vertices as quaternions, they form a group under quaternion multiplication, and this group is isomorphic to the binary tetrahedral group.

### Third stable homotopy group of spheres

The third stable homotopy group of spheres is the cyclic group of order 24.

### String theory

Critical Bosonic string theory requires 26 spacetime dimensions of target spacetime (for vanishing dilaton, at least), which can be broken down into 2 dimensions for the string worldsheet itself and 24 transversal directions for its oscillations.

This is also related to the conventional choice of normalization for the central charge of the Virasoro algebra:

$[L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12}(m^3 - m) \delta_{m+n,0} \, .$

Several classes of string theory vacua require the presence of exactly 24 branes of codimension 4 transverse to a K3-surface-fiber.

Appearances of 24 and its factors, particularly 8 and 12, are a meta-pattern in mathematics; another such meta-pattern is ADE classification. Sometimes these meta-patterns overlap.

For example, tessellating $\mathbb{R}^4$ with regular 24-cells creates the 24-cell honeycomb, in which the centers of the 24-cells are the points of the $D_4$ lattice. Meanwhile, the binary tetrahedral group corresponds to the $E_6$ Dynkin diagram (see McKay correspondence).

Families of two-dimensional conformal field theories, in which the Virasoro algebra plays a key role, are among the examples falling into an ADE classification.

## References

Blog discussion:

Discussion in relation to the Leech lattice:

• N. J. Sloane, A note on the Leech lattice as a code for the Gaussian channel. Information and Control, 46& 3 (1980). 270–272 (pdf)

• Terry Gannon, section 2.5.1 of: Moonshine Beyond the Monster, Cambridge University Press, 2006 (doi:10.1017/CBO9780511535116)

• H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, Universal optimality of the $E_8$ and Leech lattices and interpolation formulas, (arXiv:1902.05438).

Discussion in relation to the J-homomorphism: