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Color Symmetry in Plane Patterns Frank A. Farris Santa Clara - - PowerPoint PPT Presentation
Color Symmetry in Plane Patterns Frank A. Farris Santa Clara - - PowerPoint PPT Presentation
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
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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!
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Stumbled on color symmetry by accident
Rendered in Excel!
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My origin story: grating def’ns of “pattern”
“A pattern is obtained by repeating a motif… ”
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My response: Patterns are made from waves!
W aves
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SymmetryWorks software allows you to play too. Public domain software written by students at Bowdoin College (and SCU)
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Connect function-theoretic approach to color symmetry
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Read the details:
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- Classifying patterns
- Two-color symmetry
- Three-color symmetry
Connect to textiles in the real world?
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Classifying patterns
W allpaper: Translational symmetry along two independent axes
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Classify patterns by symmetries
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Too many to list!
Mirrors 4-Centers Glide reflections
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Group concept: Compose two symmetries, get a symmetry
“freedom and constraint”
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Learn how to draw a “fundamental cell”
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Complete cell diagram with…
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Group concept: a few symmetries can generate all
This group is called p4 g by the International Union of Crystallographers
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p4g can be generated by
- ne mirror, one rotation
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Two patterns have the same “type” if their symmetry groups are the same
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Wallpaper surprise:
Exactly 17 types
p4g
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Chalk Slam at Carleton
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See them everywhere!
pmg pattern from The Shining
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Construct wallpaper functions
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SymmetryWorks screenshot
Written by students at Bowdoin College, based on software by SCU students. Handout available
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A few favorites: Minimalist
Recognize p4g?
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A few favorites: Fanciful
Machine learning project: Find patterns that humans will like
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Color-reversing (or 2-color) symmetry
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Requires a color-reversing wheel
And the right waveforms
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A break for algebra: Suppose
Homomorphism
Color gp
- Sym. gp
How many ways?
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17 symmetry types 46 color-reversing types
Actual symmetry group: p3 Color-reversing or –preserving symmetries: p6 type: p6/ p3
Color-reversing half turn
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Actual symmetry group: pmg Color-reversing or –preserving symmetries: cmm type: cmm/ pmg
Color-reversing half turn
Color black counts as 0
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THE GEOMETRICAL BASIS OF PATTERN DESIGN. Part IV: Counterchange Symmetry in Plane Patterns, by H. J. Woods
Figures from Grünbaum and Shephard, Tilings and Patterns Ad hoc notations like p4g[ 2] _2 Journal of Textiles (Manchester) 1936
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Figures from appendix of Creating Symmetry Types p4m/ cmm and p4g/ cmm
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Recipes for 63 types
These recipes are not encoded in current software
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Three-color (color-turning) symmetry
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Three-color (color-turning) symmetry
Diagonal 1/ 3 translations are color-turning
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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
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Connect to the non-digital world?
“Woven” polyhedral shapes
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Color gp O
- Sym. gp
T 24 Bands
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Aside: Polyhedral Sampler from 2015
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Icos from five tetrahedra 60 bands. Does not have 5-color symmetry
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D6/ D3
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