Course:Harris, Fall 07: Diary Week 3

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  • 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


  • 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


  • 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