Periodic and aperiodic tilings

Abstract

The diversity of the phenomena of nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment. Johannes Kepler (1571–1630) In this chapter we build upon the concepts of 2-D Bravais lattices and 2-D plane groups, described in Chapters 3 and 9, to introduce the mathematics, nomenclature, and classification schemes of 2-D periodic tilings . Since quasi-periodic and aperiodic tilings , such as the Penrose tile , have become important in crystallography (e.g., quasicrystallography ), we describe these important tilings in this chapter. A detailed discussion of quasicrystallography is left for Chapter 19. Finally, we discuss the construction of 3-D structures from the stacking of 2-D tiles, and the tiling of an n -D space with polyhedra (in 3-D) or polytopes (in higher-dimensional spaces, i.e., n > 3). 2-D plane tilings In the mathematical literature, a tiling is synonymous with a tessellation . The theory of tilings is rich, and we will introduce several concepts that are useful for the classification of crystal structures. More detailed information can be found in the book Tilings and Patterns (Grunbaum and Shepard, 1987), which is an authoritative treatment of this subject. An older text, Mathematical Models (Cundy and Rollet, 1952), also covers this topic, and played a role in the definition of the Frank–Kasper phases , which will be discussed in Chapter 18. The book Quasicrystals and Geometry (Senechal, 1995), offers an excellent review of aperiodic tilings and quasicrystals.