Thursday, September 25
Collect Project Proposals and 1 or 2
portfolio sample entries.
ISOMETRIES: Rigid Motions in (or about)
the plane. Also called "Isometries"
Orientation preserving
Translations
Rotations
Orientation reversing
Reflections
Glide reflections
Classification of Isometries
Video : Isometries
The video introduced the four isometries we have discussed:
reflections, rotations, translations, and glide reflections.
It was shown 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).
Wingeometry demonstration for reflection-
one and two reflections
What about 3 reflections?
Every plane isometry is the
product of at most three reflections.
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.
Two reflections = rotation or translation.
Three reflections = reflection or glide reflection
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 tilings180 degree Rotations
Translations
Space: How do we understand objects in space?
How can the Flatlander experience the sphere and space?
Cross sections
shadows
fold downs- flattened figures
analogue... point... line.... polygon.... polyhedron......
- Some Issues we'll consider in space:
-
Platonic (regular convex polyhedra) Solids
-
Why are there only 5?
-
Regular polygons around a vertex.
-
All vertices are "the same".
-
Symmetries (Isometries) in the plane compared to those in space- an introduction:
-
Translations
-
Rotations: Center point - central axis
-
Reflection : across line - across plane
-
Symmetries of the cube:
-
Rotations
-
reflections
-
rotation- reflection
- Isometries in space: products of reflections in space:
-
Rotations and translations
-
Applications to dance