Quanta in Quasicrystals, with Dual Harmonics

Relevance: Condensed matter often spawned new general physics (Fermi level, fermions, bosons etc.) Bragg’s law depends on harmony in integral order. Diffraction in Quasicrystals requires simultaneous harmony between periodic probes and geometric series diffraction. Notice that quanta are confined by harmonic states (Schrödinger’s eigenstates). Fact: Quasicrystal diffraction [1] is in irrational, aperiodic and geometric order, with icosahedral point group symmetry. The indices due to 4-D hierarchic icosahedra are separable into natural numbers with irrational residues. The latter cause metric stretch that enables commensurate harmony in both linear and geometric space. The stretch causes translational symmetry, both long range and short, in the quasi-Bloch-wave probe. Numerical quasi-structure-factor simulations exactly match both Fibonacci series analysis and verification by quasi-lattice parameter measurement. Quasicrystals teach how the momentum quantization results from the dual harmonics. Corollary: Dimensions should not be multiplied without necessity; nor fields; nor quanta (of gravity). [1] Journal of Modern Physics, (2021) 12 1618-1632. doi: 10.4236/jmp.2021.1212096