Thursday,  September 25
Collect Project  Proposals  and   1 or 2 portfolio sample entries.
• ISOMETRIES: Rigid Motions in (or about) the plane.  Also called "Isometries"
• Orientation preserving
• Translations
• Rotations
• Orientation reversing
• Reflections
• Glide reflections

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

It was shown 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

Every plane isometry is the product of at most 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

• Using Isometries to create variations of tilings
• 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:
• Rotations
• reflections
• rotation- reflection
• Isometries in space: products of reflections in space:
• Rotations and translations
• Applications to dance