Course:Harris, Fall 07: Diary Week 3

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 (Dn)
    • without a reflection: group of symmetries is cyclic (Cn)
  • 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
• 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