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How we search for symmetry by breaking it
Rebecca Waldecker and Wilf Wilson
University of Halle-Wittenberg
Forum “Experiment!” 2019
Image credit: pixabay.com
How we search for symmetry by breaking it Rebecca Waldecker and Wilf - - PowerPoint PPT Presentation
How we search for symmetry by breaking it Rebecca Waldecker and Wilf - - PowerPoint PPT Presentation
How we search for symmetry by breaking it Rebecca Waldecker and Wilf Wilson University of Halle-Wittenberg Forum Experiment! 2019 Image credit: pixabay.com Image credit: pixabay.com Image credit: pixabay.com Image credit: pixabay.com
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Image credit: pixabay.com
Image credit: pixabay.com
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Image credit: pixabay.com Image credit: pixabay.com
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Three-Dimensional Structure of the Human Herpesvirus 8 Capsid (Journal of Virology 2000)
by Lijun Wu, Pierrette Lo, Xuekui Yu, James K. Stoops, B. Forghani, and Z. Hong Zhou.
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How many circles?
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Symmetry with permutations
The symmetries of a square
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Symmetry with permutations
1 2 4 3
The symmetries of a square
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Symmetry with permutations
1 2 4 3
The symmetries of a square
90º rotation
1 2 2 3 3 4 4 1
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Symmetry with permutations
1 2 4 3
The symmetries of a square
90º rotation
1 2 2 3 3 4 4 1
Which permutations of 1, 2, 3, 4 give symmetries?
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n! = n · (n - 1) · (n - 2) · ⋯ · 2 · 1
There are… …permutations of n numbers
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n! = n · (n - 1) · (n - 2) · ⋯ · 2 · 1
There are… …permutations of n numbers
5! = 120
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n! = n · (n - 1) · (n - 2) · ⋯ · 2 · 1
There are… …permutations of n numbers
5! = 120 15! = 1307674368000
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n! = n · (n - 1) · (n - 2) · ⋯ · 2 · 1
There are… …permutations of n numbers
5! = 120 30! = 265252859812191058636308480000… 000 15! = 1307674368000
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Graphs
1 2 3 4 5 6
Vertices (nodes) Arcs (lines) Can show relationships
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Searching with graphs
1 2 3 4 5 6
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Searching with graphs
1 2 3 4 5 6
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Searching with graphs
1 2 3 4 5 6
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Searching with graphs
1 2 3 4 5 6
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Searching with graphs
1 2 3 4 5 6
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Is this Crazy?!
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The Team
Rebecca Waldecker Wilf Wilson
University of Halle-Wittenberg
Chris Jefferson Markus Pfeiffer
University of St Andrews (Scotland)
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Paper: “Permutation group algorithms based on directed graphs”
https://arxiv.org/abs/1911.04783
Software: “GraphBacktracking” package for GAP
https://github.com/peal/GraphBacktracking
Publications
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Experiments 1
Old New
Difficulty of problems
Amount of searching needed (more is worse)
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Experiments 2
Old New
Amount of searching needed (more is worse)
Problem instance, sorted by difficulty
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What we learned Our biggest challenges What’s next?
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What we learned Our biggest challenges What’s next?
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What we learned Our biggest challenges What’s next?
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What we learned Our biggest challenges What’s next?
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