Tuesday,  October 4

Review of Discussion So Far on Classification of Isometries
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).

We saw this also with a Wingeometry demonstration for reflection- one and two reflections

The video shows that
 Any plane isometry  is either a reflection or  the product of two or three reflections.
Two reflections = rotation or translation.
What about 3 reflections? 

Three reflections = reflection or glide reflection
Visual Proof discussion from Math 371 (HSU Geometry Course): Key idea- The product of two reflections is "flexible."

Comment on Symmetry  in Music and Sound: (another dimension?)

Translation   .... by an octave, relative position-- chords
Rotation: thirteen notes in "chromatic" scale
Glide Reflection
Visualizations of music - Examples:
Notation: Conventional music notation:  The mozart viewer

Windows media player.

Space: How do we understand objects in space?
How can the Flatlander experience the sphere and space?
Pick up templates to make Platonic solid models for next class!

The simplest three dimensional figure has 4 points not all in the same plane: three point determine a plane- so a fourth point not in that plane will need "space" to make sense. These four points determine a tetrahedron.

Cross sections: Look at the tetrahedron with cross sections : Triangles, what if the tetrahedron starts through Flatland with an edge first?

Shadows:  A sphere might cast a circular shadow, but more typical the sphere's shadow is a cone in space and thus casts an elliptical shadow!
We considered how the tetrahedron might case shadows. Sometimes a triangle, sometimes a quadrilateral.