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.