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).

**
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.**

Proof: Click here!

What about 3 reflections?

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

Comment on

Translation .... by an octave, relative position-- chords

Reflection

Rotation: thirteen notes in "chromatic" scale

Glide Reflection

Visualizations of music - Examples:

Notation: Conventional music notation: The mozart viewer

Windows media player.

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.