The immortaluty of Platonic solids
The paper examines the significance of Platonic Solids in different parts of modern science.
The book contains the reports of the member of the congress from the different countres. They consider the idea of the symmety in the science and in the art.
The symmetry nature of the appearance of specific surface (edge) states at the boundaries of low_dimension structures with the symmetry of ribbons (borders) invariant with respect to time reversal is discussed. Symmetry reasons for the stability of such states against the elastic scattering from nonmagnetic impurities have been revealed.
A new type of massless Dirac fermions in crystalline three_dimensional topological insulators (three_dimen_sional two_dimensional situation) has been predicted. The spectrum has fourfold degeneracy at the top of the two_dimensional Brillouin zone (M point) and twofold degeneracy near the M point. Crystal symmetry along with the time reversal invariance in three_dimensional topological insulators allows fourfold degenerate Dirac cones, which are absent in the classification of topological features in R._J. Slager et al., Nat. Phys. 9, 98 (2013). The Hamiltonian in the cited work does not contain Dirac singularities with more than twofold degeneracy. For this reason, the corresponding topological classification is incomplete. The longitudinal magnetic field in the spinless case holds the massless dispersion law of fermions and does not lift fourfold degeneracy. In the spinor case, the magnetic field lifts fourfold degeneracy, holding only twofold degeneracy, and results in the appearance of a band gap in the spectrum of fermions.
The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established.