$4$-crystals and quasicrystals

$4$-crystals are periodic material structures in the $4$-dimensional space-time. The classical crystallographic relations are generalized to this case. The quasicrystals are solids whose X-ray photograph and electronogram contain regular pentagons. It is shown that the quasicrystals obtained experimentally by Shechtman et al. are $4$-crystals.