Thursday, February 24
Prescreening comment on President Summers (Harvard), women and science: President's letter to faculty.
Association for Women in Mathematics's response to Harvard President Lawrence Summers' remarks
Letter from the President of the Anita Borg Institute for Women and Technology: Telle Whitney responds to Lawrence Summers.
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
Reflection
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!
Recall assignment: Make a torus with 2 and 1 piece!
Cross sections: Look at the octahedron with cross sections : squares,
rectangles, triangles and hexagons depending on how the octahedron passes
through the plane.
Shadows: Recall our previous class activity when we considered how the octhedron might case shadows.
Fold downs flattened figures: Consider how the cube can be
assembled from folded down squares in two different configurations: a
cross or a "zigzag."
What does a folded down flattened torus look like?
A rectangle with opposite sides resulting from cutting the torus open making
a cylinder and then cutting the cylinder along its length.
A torus
analogue... point... line.... polygon.... polyhedron......
 Some Issues we'll consider in the plane and in space:
 Solids and Surfaces: Geometry and Topology [Classification]
 Transformations: Rigid, Projective, "continuous"
 Invariants examples: Measurements, qualities, counting
 Isometries in space:
 products of reflections
in space:
 Rotations and translations
 Applications to dance
 Polyhedra and symmetry.
 Historical Note on Kepler.
 Platonic (regular convex polyhedra) and Archimedean
(semiregular convex) Solids on the
web!


Why are there only 5?
 Look at the possible ways to put a single
regular polygon together with more of the same to make a spatial "cap"
about a single vertex. This involves equilateral triangles (3,4, or 5),
squares (3) or pentagons (3).
 This shows that there were at most five vertex caps possible. These actually do work to make
 Regular polygons around a vertex.
 All vertices are "the same"
 Activity: Counting vertices, edges, and faces for the platonic solids to become more familiar with them.