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LSP operations
▶ periodic tiling of the plane ▶ barycentric subdivision ▶ 3 mirror axes
Construction of Local Symmetry Preserving Operations Pieter - - PowerPoint PPT Presentation
Construction of Local Symmetry Preserving Operations Pieter - - PowerPoint PPT Presentation
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
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LSP operations
▶ periodic tiling of the plane ▶ barycentric subdivision ▶ 3 mirror axes
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LSP operations
▶ periodic tiling of the plane ▶ barycentric subdivision ▶ 3 mirror axes
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Can we construct all LSP operations?
Problems
▶ Different tilings can produce the same operation ▶ This defjnition is not convenient for computers
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Can we construct all LSP operations?
Problems
▶ Different tilings can produce the same operation ▶ This defjnition is not convenient for computers
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Decorations
▶ labeled planar graph ▶ 2-connected ▶ one outer face, 3 labeled corners ▶ inner faces are triangles ▶ extra constraints
▶ labels ▶ degrees ▶ border and corners
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Can we construct all decorations?
Problems
▶ Many constraints are diffjcult to program with ▶ Different constraints in the corners → diffjcult to extend
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Can we construct all decorations?
Problems
▶ Many constraints are diffjcult to program with ▶ Different constraints in the corners → diffjcult to extend
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Predecorations
▶ labeled planar graph ▶ connected ▶ one outer face ▶ inner faces are quadrangles ▶ degree of inner vertices > 2
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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
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Can each predecoration be completed?
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Can each predecoration be completed?
Extra condition: n0 ≤ 2, n0 + n1 + n2 ≤ 3
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Can each predecoration be completed?
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Completion
Theorem
Each decoration is the completion of a predecoration.
Remark
Not each predecoration can be completed.
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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
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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
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Extensions
1 2 3 4 5 6 7 8 9 10
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Construction
Lemma
Ordered reductions preserve the predecoration properties.
▶ connected ▶ inner faces are quadrangles ▶ degree of inner vertices > 2 ▶ n0 ≤ 2, n0 + n1 + n2 ≤ 3
Theorem
Each predecoration can be constructed from one edge with the 10 extensions.
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Construction
Lemma
Ordered reductions preserve the predecoration properties.
▶ connected ▶ inner faces are quadrangles ▶ degree of inner vertices > 2 ▶ n0 ≤ 2, n0 + n1 + n2 ≤ 3
Theorem
Each predecoration can be constructed from one edge with the 10 extensions.
SLIDE 21
Construction
Lemma
Ordered reductions preserve the predecoration properties.
▶ connected ▶ inner faces are quadrangles ▶ degree of inner vertices > 2 ▶ n0 ≤ 2, n0 + n1 + n2 ≤ 3
Theorem
Each predecoration can be constructed from one edge with the 10 extensions.
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Algorithm
▶ Construct predecorations ▶ Complete to decoration if possible ▶ Filter for the wanted infmation factor
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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