# Course:Harris, Fall 07: Diary Week 3

From EscherMath

Mon:

- finished looking at Rotational and Reflectional Symmetry Exploration
- noted presence of some translational symmetry in butterfly pattern (print 12)

- addressed general notion: classify group of symmetries in patterns or objects
- reflections (specify reflection-axis)
- rotations (specify rotation-center, order of rotation)
- translations (specify direction, amount of translation)
- glide-reflections (specify reflection-axis, amount of glide)
- example: butterfly pattern

- classes of objects whose symmetry groups we'll look at:
- objects of finite extent: rosettes
- objects of linearly infinite extent: friezes
- objects of two-dimensional infinite extent: wallpaper patterns

- looked at Celtic Art Exploration
- rosette symmetry groups have
- possibly rotational symmetry
- possibly one or more reflection symmetry
- with a reflection: group of symmetries is dihedral (D
*n*) - without a reflection: group of symmetries is cyclic (C
*n*)

- with a reflection: group of symmetries is dihedral (D
- no translations, no glide-reflections

Wed:

- a group is a collection of operations (always including the "null" operation) that
- can all be combined one with another
- can each be inverted

- claim: symmetry group for a finite pattern (rosette) cannot have parallel reflection axes nor multiple rotation centers:
- parallel reflection axes create an infinite pattern:
- looked at how that happens, essentially same as infinite images in facing mirrors

- two rotation centers create an infinite pattern:
- looked at how that happens with Rosette Exercise #13

- parallel reflection axes create an infinite pattern:
- began doing Tessellation Exploration in class

Fri:

- collected Rosette Exercises
- class finished up Tessellation Exploration
- handed out alternate explanation of Frieze Groups
- class did Border Pattern Exploration
- major goals with symmetry groups:
- given a pattern (rosette, frieze, or wallpaper), identify the symmetry group by name
- given a specific symmetry group (rosette, frieze), build up a pattern that has that as its symmetry group

- Frieze Exercises due Monday
- upcoming: field trip to the Cathedral
- cancel one day of class (likely Friday of next week)
- groups should plan to go together some time next week and search for symmetry groups