Tuesday, February 24
Comment on Project Proposals and portfolio
sample entries.
Review of Discussion So Far on Classification of Isometries
Video : Isometries
The video introduced the four isometries we have discussed:
reflections, rotations, translations, and glide reflections.
It showed 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
he video showed 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
Preview of visual Proof: Key idea The product of two reflections is "flexible."
How to figure out what isometry you have.... match features.

Preserve
Orientation 
Reverse
Orientation 
No Fixed points 
Translation 
Glide reflection 
Fixed Point(s) 
Rotation 
Reflection 
Using Isometries to create variations of tilings
Kali:
Symmetry group
180 degree Rotations
Activity for modifying tilings
Translations
Comment on Symmetry
in Music and Sound:
we discussed
Translation .... by an octave, relative position chords
Reflection
Rotation: thirteen notes in "chromatic" scale
Glide Reflection
Space: How do we understand objects
in space?
How can the Flatlander experience the sphere and space?
Try making a torus with 2 and 1 piece!
Cross sections: Look at the cube with cross sections : squares,
rectangles, triangles and hexagons depending on how the square passes
through the plane.
Shadows: Look at how the cube might case shadows that were square, rectangular or hexagonal,
Fold downs flattened figures: Consider how the cube can be
assembled from folded down squares in two different configurations: a
cross or a "zigzag."
analogue... point... line.... polygon.... polyhedron......
 Some Issues we'll consider in space:
 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.
 Isometries in space: products of reflections
in space:
 Rotations and translations
 Applications to dance