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 describing the connection between the time and the position of a planet on its orbit around the Sun. At present this equation is usually derived from Kepler’s first and second laws by means of infinitesimal calculus, but Kepler found away to obtain it using geometric properties of an ellipse and a circle. Kepler, however, examined purely mathematical problems as well. Recall, for instance, his famous conjecture on the densest sphere packing. There is still discussion on the validity of its “computer-based proof” given in [12], since it is not in integer arithmetic (only in real computer arithmetic). Kepler is regarded as one of the founders of crystallography. He was astonished by the beauty of snowflakes when he walked over the Charles Bridge in Prague. In his treatise Strena seu de nive sexangula (1611)