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Color Symmetry in Plane Patterns Frank A. Farris Santa Clara - PowerPoint PPT Presentation

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Color Symmetry in Plane Patterns Frank A. Farris Santa Clara University The symmetries of a pattern form a group A symmetry is a transformation that leaves the pattern invariant Rosettes, friezes, wallpaper My origin story: grating defns

  1. Color Symmetry in Plane Patterns Frank A. Farris Santa Clara University

  2. The symmetries of a pattern form a group A symmetry is a transformation that leaves the pattern invariant Rosettes, friezes, wallpaper

  3. My origin story: grating def’ns of “pattern” “A frieze pattern is a set of points that is invariant under… ” Set of points? A pattern is a function!

  4. Stumbled on color symmetry by accident Rendered in Excel!

  5. My origin story: grating def’ns of “pattern” “A pattern is obtained by repeating a motif… ”

  6. My response: Patterns are made from waves! W aves

  7. SymmetryWorks software allows you to play too. Public domain software written by students at Bowdoin College (and SCU)

  8. Connect function-theoretic approach to color symmetry

  9. Read the details:

  10. Classifying patterns • Two-color symmetry Connect to textiles • in the real world? Three-color symmetry •

  11. Classifying patterns W allpaper : Translational symmetry along two independent axes

  12. Classify patterns by symmetries

  13. Too many to list! Mirrors 4-Centers Glide reflections

  14. Group concept: Compose two symmetries, get a symmetry “freedom and constraint”

  15. Learn how to draw a “fundamental cell”

  16. Complete cell diagram with…

  17. Group concept: a few symmetries can generate all This group is called p4 g by the International Union of Crystallographers

  18. p4g can be generated by one mirror, one rotation

  19. Two patterns have the same “type” if their symmetry groups are the same

  20. Wallpaper surprise: Exactly 17 types p4g

  21. Chalk Slam at Carleton

  22. See them everywhere! pmg pattern from The Shining

  23. Construct wallpaper functions

  24. SymmetryWorks screenshot Written by students at Bowdoin College, based on software by SCU students. Handout available

  25. A few favorites: Minimalist Recognize p4g?

  26. A few favorites: Fanciful Machine learning project: Find patterns that humans will like

  27. Color-reversing (or 2-color) symmetry

  28. Requires a color-reversing wheel And the right waveforms

  29. A break for algebra: Suppose Color gp Sym. gp Homomorphism How many ways?

  30. 17 symmetry types 46 color-reversing types Actual symmetry group: p3 Color-reversing or –preserving symmetries: p6 type: p6/ p3 Color-reversing half turn

  31. Actual symmetry group: pmg Color-reversing or –preserving symmetries: cmm type: cmm/ pmg Color black counts as 0 Color-reversing half turn

  32. THE GEOMETRICAL BASIS OF PATTERN DESIGN. Part IV: Counterchange Symmetry in Plane Patterns, by H. J. Woods Journal of Textiles (Manchester) 1936 Figures from Grünbaum and Shephard, Tilings and Patterns Ad hoc notations like p4g[ 2] _2

  33. Figures from appendix of Creating Symmetry Types p4m/ cmm and p4g/ cmm

  34. Recipes for 63 types These recipes are not encoded in current software

  35. Three-color (color-turning) symmetry

  36. Three-color (color-turning) symmetry Diagonal 1/ 3 translations are color-turning

  37. Connect to the non-digital world? Woven Rope Friezes, with Hans Kristian Rossing in Mathematics Magazine, 1999 Physical object not invariant under mirror symmetry, but rather under half-turn symmetry in space

  38. Connect to the non-digital world? “Woven” polyhedral shapes

  40. Color gp O Sym. gp T 24 Bands

  41. Aside: Polyhedral Sampler from 2015

  42. Icos from five tetrahedra 60 bands. Does not have 5-color symmetry

  43. D6/ D3

  44. Question: Which frieze and wallpaper patterns can be woven in chain mail? Color patterns?

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