Lesson Plan, Reading and Written Assignments for Weeks 7 and 8
The readings and assignments will be in Brown, Transformational Geometry.
The plan is to finish the essentials
of transformational geometry (not that we will exhaust it) by Thanksgiving,
with applications to geometry and symmetry later.
Readings for Classes
Prerequisite reading and review, read sections 1.1., 1.2, 1.3
This beginning material on functions and transformations
should be familiar to you from prerequisite classes such as linear algebra and
calculus. Look it over and read
carefully if you need to review. The key
ideas include the concept of mapping and a transformation of the plane as a
one-to-one mapping with domain and range equal to the whole plane.
Monday, 11/7: Read sections 1.4 and
1.6.
- Definition
of line reflection
- Definition
of isometry
- Proof
that a line reflection is an isometry
- Proof
that an isometry maps a triangle to a congruent triangle.
Wednesday, 11/9: Read sections 1.6
and 1.7, also first theorem in 1.10
- Properties
of isometries: map lines to lines, circles to circles, parallels to
parallels; angles are preserved.
- Fundamental
Theorem 10. For an isometry, if the image of a triangle is known, the
image of any point can be constructed (see 1.10).
- Definition
of a rotation
- Proof
that a rotation is an isometry
- Finding
the center of a rotation given a figure and its image
Wednesday, 11/9 (487 Lab): Read
sections 1.7 and 1.8.
- Double
reflections in intersecting lines are rotations and vice versa (see
Theorem 6 and p. 29 figure 1.40).
- Double
reflections in parallel lines are translations and vice versa (see Theorem
8).
Friday, 1/11: (Veteran's Day Holiday)
Monday, 11/14: Read sections 1.8
(glide reflections) and 1.10.
- Definition
and properties of glide reflections
- Translations
and glide reflections are isometries because compositions of isometries
are isometries.
- Fundamental
Triple line reflections and solving ab = cd for transformations.
- Fundamental
Theorems 11 and 12.
Wednesday, 11/16: Read sections 2.1
and 2.4.
- Composition
of transformations in general -- order, associativity, and inverses.
- Proof of
the nature of triple line reflections (problems 17-19, proving the missing
part of Theorem 12).
- Composing
two rotations. (Theorems 23 and 24)
Wednesday, 11/16 (487 Lab): Read
sections 2.2 and 2.3.
- Practice
from 2.1 and 2.4.
- Algebra
of translations.
- Algebra
of half-turns.
Friday, 11/18: Read section 2.5.
Semi-quiz B.
- Concept
of a group.
- All
isometries form a group.
- Translations
form a group. Also rotations +
translations.
- Half-turns
do not form a group.
- Line
reflections do not form a group, nor do glide reflections.
- Semi-quiz B today.
Monday, 11/21: Read sections 1.5
(reread) and 2.7.
- Concept
of symmetries and symmetry groups.
- Examples
of finite groups.
- Examples
of frieze groups.
Wednesday, 11/23: Read supplement on symmetry
- Tessellations
- Wall-paper
groups and quilts.
- Apply
composition theorems to analyze symmetries.
Wednesday, 11/23 (487 Lab): (take-home or in lab)
- Tessellation
examples
- Symmetry
examples with 90-degrees
- Symmetry
examples with 60 or 120 degrees.
Friday, 11/25: (Thanksgiving Holiday)