Thursday,  February 17
Finish FAPP Tape on Symmetry
Using Symmetries to create variations of tilings
Kali: Symmetry group 
  • 180 degree Rotations 
  • Translations
    •  
    World of Escher
    ISOMETRIES: Rigid Motions in (or about) the plane.  Also called "Isometries"

    Orientation preserving :
    Translations
    Rotations
    Orientation reversing:
    Reflections
    Glide reflections

    Classification of Isometries
    prepare for Video : Isometries (show video on Tuesday)
    The video introduces the four isometries we have discussed:
    reflections, rotations, translations, and glide reflections.

    Show that the product of two reflections is either a rotation (if the axes of the reflection intersect)  or a translation (if the axes of the reflection are parallel).
    Wingeometry demonstration for reflection- one and two reflections
    What about 3 reflections? 

     Any plane isometry  is either a reflection or  the product of two or three reflections.

    discuss basic idea:
    Reflection is related to "perpendicular bisector" of PP'
    With a triangle the 3 vertices ABC -> A'B'C' may be related to at most 3 lines of reflection.
    Two reflections = rotation or translation.
    Three reflections = reflection or glide reflection
    How to figure out what isometry you have.... match features.
     

    		Preserve
    Orientation    		 Reverse
    Orientation
    No Fixed points Translation Glide reflection
    Fixed Point(s) Rotation Reflection


    Space: How do we understand objects in space?
    How can the Flatlander experience the sphere and space?

    Cross sections
    shadows
    fold downs- flattened figures
    analogue...  point... line.... polygon.... polyhedron......
    • Some Issues we'll consider in space:
    • Platonic (regular convex polyhedra) Solids
      • Why are there only 5?
        • Regular polygons around a vertex.
        • All vertices are "the same".
    • Symmetries (Isometries) in the plane compared to those in space- an introduction:
        • Translations
        • Rotations: Center point - central axis
        • Reflection :  across line - across plane
    • Symmetries of the cube:
    • Isometries in space: products of reflections in space:
          • Rotations and translations
          • Applications to dance