Thursday,  February 19

    Tuesday  8-10 am, HGH 105;     Wednesday 8-11am : SCIA 364 . 

ISOMETRIES: Rigid Motions in (or about) the plane.  Also called "Isometries"

Orientation preserving :

Orientation reversing:
Glide reflections

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

It shows 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.

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

Using Isometries to create variations of tilings
Kali: Symmetry group
  • 180 degree Rotations 
  • Translations

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

    Cross sections
    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