Construction of Local Symmetry Preserving Operations Pieter Goetschalckx Ghent University Department of Applied Mathematics, Computer Science and Statistics August 2017 – Computers in Scientifjc Discovery 8
▶ barycentric subdivision ▶ 3 mirror axes LSP operations ▶ periodic tiling of the plane
▶ 3 mirror axes LSP operations ▶ periodic tiling of the plane ▶ barycentric subdivision
LSP operations ▶ periodic tiling of the plane ▶ barycentric subdivision ▶ 3 mirror axes
Problems ▶ Different tilings can produce the same operation ▶ This defjnition is not convenient for computers Can we construct all LSP operations?
Can we construct all LSP operations? Problems ▶ Different tilings can produce the same operation ▶ This defjnition is not convenient for computers
Decorations ▶ labeled planar graph ▶ 2-connected ▶ one outer face, 3 labeled corners ▶ inner faces are triangles ▶ extra constraints ▶ labels ▶ degrees ▶ border and corners
Problems ▶ Many constraints are diffjcult to program with ▶ Different constraints in the corners → diffjcult to extend Can we construct all decorations?
Can we construct all decorations? Problems ▶ Many constraints are diffjcult to program with ▶ Different constraints in the corners → diffjcult to extend
Predecorations ▶ labeled planar graph ▶ connected ▶ one outer face ▶ inner faces are quadrangles ▶ degree of inner vertices > 2
Predecorations Decoration → predecoration ▶ Remove red and blue edges Predecoration → decoration ▶ Fill quadrangles with X’s ▶ Attach T’s ▶ Satisfy contraints in corners ▶ Remove cutvertices ▶ not unique
Can each predecoration be completed?
Can each predecoration be completed? Extra condition: n 0 ≤ 2 , n 0 + n 1 + n 2 ≤ 3
Can each predecoration be completed?
Completion Theorem Each decoration is the completion of a predecoration. Remark Not each predecoration can be completed.
Canonical construction path method ▶ 10 extensions/reductions ▶ Start with a single edge ▶ Apply extensions ▶ Check if canonical ▶ Find canonical orbit of reductions ▶ Check if last extension is inverse Can we construct all predecorations?
Can we construct all predecorations? Canonical construction path method ▶ 10 extensions/reductions ▶ Start with a single edge ▶ Apply extensions ▶ Check if canonical ▶ Find canonical orbit of reductions ▶ Check if last extension is inverse
Extensions 1 2 3 4 5 6 7 8 9 10
▶ connected ▶ inner faces are quadrangles ▶ degree of inner vertices > 2 ▶ n 0 ≤ 2 , n 0 + n 1 + n 2 ≤ 3 Theorem Each predecoration can be constructed from one edge with the 10 extensions. Construction Lemma Ordered reductions preserve the predecoration properties.
Theorem Each predecoration can be constructed from one edge with the 10 extensions. Construction Lemma Ordered reductions preserve the predecoration properties. ▶ connected ▶ inner faces are quadrangles ▶ degree of inner vertices > 2 ▶ n 0 ≤ 2 , n 0 + n 1 + n 2 ≤ 3
Construction Lemma Ordered reductions preserve the predecoration properties. ▶ connected ▶ inner faces are quadrangles ▶ degree of inner vertices > 2 ▶ n 0 ≤ 2 , n 0 + n 1 + n 2 ≤ 3 Theorem Each predecoration can be constructed from one edge with the 10 extensions.
Algorithm ▶ Construct predecorations ▶ Complete to decoration if possible ▶ Filter for the wanted infmation factor
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Questions?
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