Classical Lattices: $\mathbb{Z}^n$, $A_n$, $D_n$, $E_8$, $\Lambda_{24}$

The integer lattice is $$\mathbb{Z}^n \;=\; \{(u_1, \ldots, u_n) : u_i \in \mathbb{Z}\}.$$ Its generator matrix is $\mathbf{I}_n$, so $V(\mathbb{Z}^n) = 1$, $\rho(\mathbb{Z}^n) = 1/2$, $R(\mathbb{Z}^n) = \sqrt{n}/2$, and $K(\mathbb{Z}^n) = 2 n$ (the $\pm \mathbf{e}_i$). The kissing number grows linearly with dimension; the packing density $$\Delta(\mathbb{Z}^n) \;=\; \frac{V_n}{2^n} \;=\; \frac{\pi^{n/2}} {2^n \Gamma(n/2 + 1)}$$ tends to zero very quickly with $n$: $\Delta(\mathbb{Z}^1) = 1$, $\Delta(\mathbb{Z}^2) \approx 0.785$, $\Delta(\mathbb{Z}^8) \approx 0.0159$, $\Delta(\mathbb{Z}^{24}) \approx 3.2 \times 10^{-7}$. The cube is the benchmark every other lattice beats.

The $A_n$ root lattice lives in the hyperplane of $\mathbb{R}^{n+1}$ of vectors with integer coordinates summing to zero: $$A_n \;=\; \{(u_0, u_1, \ldots, u_n) \in \mathbb{Z}^{n+1} : u_0 + u_1 + \cdots + u_n = 0\}.$$ It has minimum-norm vectors of squared length $2$ (permutations of $(1, -1, 0, \ldots, 0)$), kissing number $K(A_n) = n(n+1)$, and fundamental volume $V(A_n) = \sqrt{n+1}$. The 2D case $A_2$ is the hexagonal lattice discussed in s01–s02 (after a rotation into $\mathbb{R}^2$), which is the densest 2D lattice.

The checkerboard lattice is $$D_n \;=\; \{\mathbf{u} \in \mathbb{Z}^n : u_1 + u_2 + \cdots + u_n \text{ is even}\},$$ i.e., the sublattice of $\mathbb{Z}^n$ consisting of "even-sum" vectors. It has index $2$ in $\mathbb{Z}^n$, so $V(D_n) = 2$. The minimum-norm vectors are the $2 \cdot n (n-1)$ permutations and sign choices of $(\pm 1, \pm 1, 0, \ldots, 0)$ with squared length $2$: $K(D_n) = 2 n (n-1)$. Packing radius $\rho(D_n) = 1/\sqrt{2}$, covering radius $R(D_n) = 1$ for $n \ge 4$ (equal to the half-diagonal of the "deep hole").

$D_n$ is the densest lattice in dimensions $n = 3, 4, 5$. In $n = 3$ it is the face-centred cubic (FCC) packing, long conjectured optimal among all 3D packings and proved by Hales in 1998 (Kepler conjecture).

The $E_8$ lattice (Gosset, Korkine–Zolotarev, Leech) is the densest 8D lattice. One construction is $E_8 = D_8^+$, i.e., $$E_8 \;=\; D_8 \;\cup\; \bigl(D_8 + (\tfrac12, \tfrac12, \ldots, \tfrac12)\bigr).$$ Equivalently, $E_8$ consists of all $\mathbf{u} \in \mathbb{Z}^8 \cup (\mathbb{Z} + \tfrac12)^8$ with coordinate sum an even integer. It has $V(E_8) = 1$, minimum-norm squared $2$, kissing number $K(E_8) = 240$, and packing density $\Delta(E_8) = \pi^4 / 384 \approx 0.2537$.

$E_8$ is self-dual and even (all squared norms are even integers). In 2017 Viazovska proved that $E_8$ is the densest sphere packing in $\mathbb{R}^8$ — lattice or otherwise — closing a fifty-year-old conjecture. Her proof constructs a modular form that witnesses the bound using the linear-programming method of Cohn and Elkies.

The Leech lattice $\Lambda_{24} \subset \mathbb{R}^{24}$ is the densest 24D lattice (Conway–Sloane attribute this folklore to Leech, $1967$) and was proved optimal among all 24D sphere packings by Cohn, Kumar, Miller, Radchenko, and Viazovska in $2017$. It has $V(\Lambda_{24}) = 1$ (self-dual), minimum-norm squared $4$ (so $d_{\min} = 2$ with a conventional normalisation), kissing number $K(\Lambda_{24}) = 196560$, and packing density $\Delta(\Lambda_{24}) = V_{24} \approx 0.00193$ (equivalently, center density $\delta = 1$).

One construction: let $\mathcal{C}_{24}$ be the $[24, 12, 8]$ extended binary Golay code. Then $$\Lambda_{24} \;=\; \bigl\{\mathbf{v} \in (2\mathbb{Z} + \mathcal{C}_{24}) \cup (2\mathbb{Z} + \mathcal{C}_{24} + \mathbf{1}) : \mathbf{v} \equiv 0 \pmod{4} \text{ or shifted}\bigr\} / \sqrt{8}$$ — up to the exact normalization, the Leech lattice is built by stacking Golay-code cosets in 24D. Conway–Sloane Ch. 12 gives several other constructions (via $A_1^{24}$, via $E_8^3$, via the Niemeier lattices).

The Coxeter–Todd lattice $K_{12} \subset \mathbb{R}^{12}$ is the densest known $12$-D lattice (also believed to be the densest $12$-D sphere packing, but unlike $E_8$ and $\Lambda_{24}$ this is not proved). It has $V(K_{12}) = 3^3 = 27$ (up to scale), minimum-norm squared $4$, kissing number $K(K_{12}) = 756$, and center density $\delta \approx 0.037$. It can be constructed as a ternary Golay code over $\mathbb{Z}_3$ lifted into $\mathbb{R}^{12}$.

In the unsolved dimensions $n = 1, 2, 3, 4, 5, 8, 24$ the optimal packing is known (Hales for $n=3$, Gauss for $n=2$, Viazovska and collaborators for $n=8, 24$, trivial for $n=1$, and $n=4, 5$ known via $D_4, D_5$). In every other dimension including $12$, the conjecture remains open.