Figure 5. Lattice and non-lattice solutions in 3D.

(A) Stacking of spheres as in an $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice. In this densest lattice in 3D, each sphere touches 12 other spheres and there are four different planar hexagonal lattices through each node. (B) Over a layer of hexagonally arranged spheres centered at γ₀ (in black) one can put another hexagonal layer by starting from one of six locations, two of which are highlighted, γ₁ and γ₂. (C) If one arranges the hexagonal layers according to the sequence (…,γ₁, γ₀, γ₂,…) one obtains the $\mathcal{F}\mathcal{C}\mathcal{C}$. Note that spheres in layer I are not aligned with those in layer III. (D) Arranging the hexagonal layers following the sequence (…,γ₀, γ₁, γ₀,…) leads to the hexagonal close packing $\mathcal{HCP}$. Again, each sphere touches 12 other spheres. However, there is only one plane through each node for which the arrangement of the centers of the spheres is a regular hexagonal lattice. This packing has the same packing ratio as the $\mathcal{F}\mathcal{C}\mathcal{C}$, but is not a lattice. (E) $\text{tr}{\mathit{J}}_{\mathcal{L}}$ for bump-function Ω with $\mathcal{L}=\mathcal{F}\mathcal{C}\mathcal{C}$ and $\mathcal{HCP}$ for various parameter combinations θ₁ and θ₂; θ₁ modulates the decay and θ₂ the support. The two packings have the same packing ratio and for this tuning curve also provide identical spatial resolution. FI: Fisher information.

DOI: http://dx.doi.org/10.7554/eLife.05979.009