When reading about close-packing using spheres of the same size, I found that Gauss had already proved the highest average density achievable by a lattice packing being $\pi/3\sqrt{3}$. Then I came across a statement on the related Wikipedia page that says
The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres ... Highest density is known only for 1, 2, 3, 8, and 24 dimensions.
Then I saw another geometry-related article on the Hermite constant, which determines how long a shortest element of a lattice in Euclidean space can be. It says
The Hermite constant is known in dimensions 1–8 and 24.
Which got me thinking, is dimension 24 (and possibly dimension 8) mathematically special? Why does it appear seemingly out of nowhere in the result of proofs, when intermediate dimensions like $18, 20, 23$ etc. still aren't accounted for yet?