Five-fold symmetry in crystalline quasicrystal lattices

Five-fold symmetry in crystalline quasicrystal lattices.

To demonstrate that crystallographic methods can be applied to index and interpret diffraction patterns from well-ordered quasicrystals that display non-crystallographic 5-fold symmetry, we have characterized the properties of a series of periodic two-dimensional lattices built from pentagons, called Fibonacci pentilings, which resemble aperiodic Penrose tilings. The computed diffraction patter...

Percolation on dual lattices with k-fold symmetry

Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p = 1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang’s argument requires only 2-fold (...

Natural quasicrystal with decagonal symmetry

We report the first occurrence of a natural quasicrystal with decagonal symmetry. The quasicrystal, with composition Al71Ni24Fe5, was discovered in the Khatyrka meteorite, a recently described CV3 carbonaceous chondrite. Icosahedrite, Al63Cu24Fe13, the first natural quasicrystal to be identified, was found in the same meteorite. The new quasicrystal was found associated with steinhardtite (Al38...

Bose-Einstein Condensates in Optical Quasicrystal Lattices

Is There a Crystal Lattice Possessing Five-Fold Symmetry?

Kepler’s Mosaics TheGermanscholar Johannes Kepler (1571–1630), who worked as a court mathematician for the Holy Roman Emperor Rudolf II in Prague, is well known especially for his astronomical discoveries.We still today admire his ability to obtain his three laws of planetary motion from data analysis of Tycho Brahe’s observations of Mars. Another outstanding discovery is Kepler’s equation desc...