I'm completely not an expert in this area. During investigating of the 3D permutohedral lattice $A^\ast_3$ and body-centered cubic (BCC) lattice, I have found that they seem to share many properties, but I can't find a source directly stating they're equivalent. Can I safely claim their equivalence? Any insights or references would be appreciated.
For example:
The voronoi cell of $A^\ast_3$ is bitruncated octahedrons and it's the same for BCC lattice.
- [1] Baek, Jongmin, and Andrew Adams. "Some useful properties of the permutohedral lattice for Gaussian filtering." other words 10.1 (2009): 0.
- [2] Entezari, Alireza, Dimitri Van De Ville, and Torsten Moller. "Practical box splines for reconstruction on the body centered cubic lattice." IEEE Transactions on Visualization and Computer Graphics 14.2 (2008): 313-328.
I have found a source refering to the BCC lattice as $D^\ast_3$; And, I have found the wikipedia claiming $A_3=D_3$ (can this deduce that $A^\ast_3=D^\ast_3$ ?)
Yes, $A_3^*$ is the BCC lattice. See e. g. Sphere Packings, Lattices and Groups by Conway and Sloane, paragraph 6.7; or, the more detailed paper by Baek et al. at doi:10.1007/s10851-012-0379-2. Also, yes, if two lattices are equal (resp. similar), then their duals are surely equal (resp. similar).