Thursday, February 24
Pre-screening 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 "zig-zag."
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.
