That density is the volume density. For most applications, you care about the density in terms of number of hyperspheres instead. The largest hypersphere that f

That density is the volume density. For most applications, you care about the density in terms of number of hyperspheres instead. The largest hypersphere that fits inside an n-dimensional unit hypercube has radius 1/2 and volume π^(n/2) (1/2)^n / Γ(n/2 + 1), https://en.wikipedia.org/wiki/Volume_of_an_n-ball#Closed_form https://en.wikipedia.org/wiki/Volume_of_an_n-ball#Closed_for... so the number of hyperspheres in a given volume scales as n^2 Γ(n/2 + 1) / π^(n/2). Of course the dominant term is the factorial growth of the Gamma function, so even without the recent improvement by Klartag, using more dimensions to encode multiple values simultaneously was already preferable.