Tuesday, October 4
Discussion So Far on Classification of 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
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
Visualizations of music - Examples:
Notation: Conventional music notation:
The mozart viewer
Windows media player.
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
Cross sections: Look at the tetrahedron with cross sections :
Triangles, what if the tetrahedron starts through Flatland with an edge
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
We considered how the tetrahedron might case shadows. Sometimes a triangle, sometimes a quadrilateral.