Cubic Fuzzy Precision: Tau-scaled Fractals from Plain-Vanilla Pentagrids

James F. Hausman and Fabien Vienne
Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture (2011)
Pages 589–592 Short Papers


The discoveries of Penrose tiles, quasicrystals, and Penrose/pentagrid duality have breathed new life into 5-fold symmetries. But in the quest for aperiodicity, cases the layman would call exotic are deemed ``regular'' and the plain-vanilla cases are deemed ``singular'', even ``exceptionally singular'', and have been left largely unexplored (paradoxically, they might be the most computationally challenging [1]). ``Fuzzy precision'' uses a mechanical approach to explore the case of perfectly periodic grids superposed on an ``exceptionally singular'' origin to find surprisingly rich metastructure in the patterns that emerge from the ranked near-misses of incommensurable integer/irrational line-crossings. ``Cubic fuzzy precision'' extends the explorations to 3D. Since these are in fact limit cases of the periodic/aperiodic landscape, perhaps they can aid in broader generalizations.