In Sphere Packings, Lattices and Groups, Conway and Sloane explore laminated lattices. If we let $X_d$ be the set of $d$-dimensional Euclidean lattices where every pair of points are separated by distance at least $2$ (so as to induce packings of unit spheres), then the sets of laminated lattices are defined as follows:

  • $\mathcal{L}_0 = \{ \Lambda_0 \}$ contains the one-point lattice.
  • $\mathcal{L}_{n+1}$ is the subset of $\{ \Lambda_{n+1} \in X_{n+1} : \exists \Lambda_n \in \mathcal{L}_n \textrm{ isometrically embedded in } \Lambda_{n+1} \}$ consisting of the lattices of maximal density (minimal determinant).

Let $\lambda_n$ be the determinant of the lattices $\Lambda_n \in \mathcal{L}_n$. Then, for $0 \leq n \leq 24$, Conway and Sloane prove the following remarkable facts:

  • $\lambda_{24+n} = 2^{-n} \lambda_n$ for all $0 \leq n \leq 24$
  • $\lambda_{24-n} = \lambda_n$ for all $0 \leq n \leq 24$

If we define $f(n) := \log_2 \lambda_n + \frac{1}{96} n(24-n)$ then these identities become, respectively:

  • $f(24 + n) = f(n)$ for all $0 \leq n \leq 24$
  • $f(24 - n) = f(n)$ for all $0 \leq n \leq 24$

Does the first identity generalise to all $n \in \mathbb{N}$? This would be equivalent to $f(n)$ being periodic with period 24 (and, by the second identity, palindromic on its period).


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