Introduction to the Mathematics of Quasicrystals by Marko V. Jaric

By Marko V. Jaric

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Models of these rhombohedra can be made by folding up these nets. (Fig. 23, top right) and the 6-cube onto a triacontahedron (Fig. 28). The rhombic dodecahedron can be partitioned into four congruent rhom­ bohedra, and the triacontahedron into twenty which fall into two congruence classes. 3. 4 Forcing Transitivity In this section we will briefly discuss some of the ways in which local order in tilings of the plane (and of three dimensional space) determines the tran­ sitivity properties of the tiling.

4 Duals of Multigrids The work of N . G. 2) has led to the discovery of an interesting new class of tilings of the plane by r h o m b s and of space by r h o m b o h e d r a . We will discuss the planar case; the three dimensional case is completely analogous. Following de Bruijn, we define a grid to be an infinite family of equispaced parallel lines. To such a grid we associate a unit vector χ normal to the grid lines; the vector determines positive and negative directions for the grid (Fig. 25).

With the lattices can act on them, we conclude that there are fourteen distinct space lattices. 3 Dissection Tilings Like " d u a l , " the word " d i s s e c t i o n " is used in tiling theory in at least three different ways: 1. The simplest way to create a new tiling from a given one is to subdivide each tile into a finite number of pieces. The result of this dissection can be called a dissection tiling. The dissection tilings of greatest interest are those which are monohedral; two examples are shown in Fig.

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