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).

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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.

Proof: Click here!

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

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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

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Using Isometries to create variations of tilings
Kali: Symmetry group 

180 degree Rotations 

Translations

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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......

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• 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:

• Rotations
• reflections
• rotation- reflection

• Isometries in space: products of reflections in space:

• Rotations and translations>
• Applications to dance