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!