Post on 31-Dec-2015
Color-reversing Symmetry
Classifyingcolor-reversing symmetry
Actual symmetry group: p4
Do the color-reversing symmetries form a group?
Color group consists of all symmetries and color-reversing symmetries.
We call the pattern type p4g/p4
Here the color group is p4g
Another example
Actual symmetry group: p3
Color group: p6
type: p6/p3
Color-reversing half turn
Negating Frogs
p4g/cmm
What types are possible?Answer uses the function nature of patterns.
Average wave over 120 degree rotations:
Chalkboard work on algebra of functions with color-reversing symmetries
Counting types of color-reversing symmetry
All color-reversing symmetries generated by a single one composed with ordinary symmetries.
G is the kernel of the color homomorphism.
Therefore, the symmetry group G is a normal subgroup of the color group Gc of index 2.
Case study: Symmetry group p2mg
p2mg/p11g
Case study: Symmetry group p2mgp2mg/p211
p2mg/p1m1
There are 17 types
p2mm/p2mg
Similar analysis for wallpaper
How many (non-equivalent) homomorphisms from each wallpaper group to the group ?
There are 46! (One of each type in my book.)
Result first appeared, with pictures, in The Journal of the Textile Institute (Manchester). H.J. Woods, 1936
Recipes for 63 types
It’s fun to experiment, once these recipes are encoded in software
Escher knew how to make these!
pg/p1
Escher knew how to make these!
pmg/pmg
Only one case where nomenclature fails: pm can be a subgroup of itself in two
different ways
Yet another twist: color-turning symmetry
p3/3 p1
How did Escher do this?
p3/3 p1
We’ve only scratched the surface!