PLANAR PATTERNS WITH FIVEFOLD SYMMETRY AS SECTIONS OF PERIODIC STRUCTURES IN 4-SPACE

Two quasiperiodic planar patterns with fivefold symmetry are derived from the root lattice A₄ in 4-space. A detailed analysis of the geometry of the A₄ Voronoi complex and its dual complex is presented with special emphasis on fivefold symmetry. By means of the general dualization method, 2D patterns are obtained, one with triangular tiles and a second which turns out to be the well-known Penrose pattern. The vertex configurations and their relative frequencies, the deflation rules, and the Fourier properties of these patterns are worked out in the framework of the dualization method and Klotz construction.