Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

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Core and Variable Vertices

Unconstrained application of to the starting point x ₀ leads to the local configurations of points discussed in Section 3.3. It is useful to classify the vertices in the limit point set based on whether or not a given vertex has a neighbor that is too close to it along a 3-fold or 5-fold direction. This classification divides the vertices into core and variable vertices. The core vertices are in every possible tiling formed by these inflation rules and are so named because they form the center core of the AD that does not intersect with other ADs when translated by forbidden separations. Some core vertices exist in local configurations that can be tiled irrespective of the rest of the tiling. For instance, about the global center of icosahedral symmetry of any of these tilings, there is a core of 840 tiles ( = 7 x 120 ) where each vertex is a core vertex. Variable vertices may further be classified according to how many too-short edges they share with other vertices, as was done in Section 3.3. The AD arising from Equation  3.10 is shown along with the location of the variable vertices in perp space in Figure  3.15 .

Figure 3.15: The AD when is omitted showing the groups of variable vertices. Vertices with a single too-short 3-fold edge are teal, while those with 1 5-fold edge are red. Blue have a 5- and 3-fold edge, and black have 2 3-fold edges. Finally, green, the least frequent, have 1 5-fold and 2 3-fold edges.

The core vertices act as a scaffolding on which a tiling corresponding to a particular choice of inflation rules can be built. Given the set of core vertices, a tiling can be constructed around it by cataloging the separations between core vertices, and such a construction constitutes another way to construct tilings. Once again, the variability in the inflation rules arises when an inflation of a long 3- or 5-fold edge must be subdivided LS or SL. For a given vertex set containing both core and variable vertices, the τ² inflation of the set will produce a new set of core vertices containing no variable vertex. The τ² inflation of every variable vertex is a core vertex. In perp space, this statement reflects the fact that AD shrunk by a factor of τ² is entirely contained in the original-sized AD. That is not to say that the core vertices are a τ^-2 copy of the AD shown in Figure  3.15 , however, for the subdivision of long 5- and 3-fold edges in para space L → LSL creates new core vertices from a given set in para and perp space.

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Next: Conclusions Up: An Ensemble of Quasiperiodic Previous: The Gluing Condition and

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