The Kepler tiling as the oldest complex surface structure in history: X-ray structure analysis of a two-dimensional oxide quasicrystal approximant

We have carried out a surface X-ray diffraction (SXRD) analysis of the approximant structure related to the recently discovered two-dimensional dodecagonal oxidic quasicrystal. The structure is characterized by the 32.4.3.4 Archimedean tiling, first described by Kepler in 1619. The tiling network is related to titanium atoms observed as protrusions in scanning tunneling microscopy images. All four titanium atoms within one two-dimensional unit cell (a0=13.1 Å, b0=12.9 Å, γ=90.5°) are surrounded by three oxygen atoms. The TiO3 units are separated by barium atoms. The total stoichiometry is given by Ba4 Ti4 O10.