Tuesday,  February 22
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
Activity: Octahedron Framework 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