Mathematical Models and Experiments on the Acoustic Properties of Granular Packing Structures (Measurement of Tortuosity in Hexagonal Close-Packed and Face-Centered Cubic Lattices)

In this study, the sound absorption characteristics of hexagonal close-packed and face-centered cubic lattices were estimated by theoretical analysis. Propagation constants and characteristic impedances were obtained by dividing each structure into elements perpendicular to the incident direction of sound waves and by approximating each element to a clearance between two parallel planes. Consequently, the propagation constant and the characteristic impedance were treated as a one-dimensional transfer matrix in the propagation of sound waves, and the normal incident sound absorption coefficient was calculated by the transfer matrix method. The theoretical value of the sound absorption coefficient was derived by using the effective density applied to the measured tortuosity. As a result, the theoretical value was becoming closer to the measured value. Therefore, the measured tortuosity is reasonable.

Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns

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