Probable nature of higher-dimensional symmetries

Lattices abound in nature from the crystal structure of minerals to the honey-comb organization of 10 ommatidia in the compound eye of insects. These arrangements provide solutions for optimal pack11 ings, efficient resource distribution and cryptographic protocols. Do lattices also play a role in how 12 the brain represents information? We focus on higher-dimensional stimulus domains, with partic13 ular emphasis on neural representations of physical space, and derive which neuronal lattice codes 14 maximize spatial resolution. For mammals navigating on a surface, we show that the hexagonal 15 activity-patterns of grid cells are optimal. For species that move freely in 3D a face-centered cubic lat16 tice is best. This prediction could be tested experimentally in flying bats, arboreal monkeys, or marine 17 mammals. More generally, our theory suggests that the brain encodes higher-dimensional sensory or 18 cognitive variables with populations of grid-cell-like neurons whose activity patterns exhibit lattice 19 structures at multiple, nested scales. 20