Martin Flashman's Courses Math 103 Summer, 2002
08JUL02 to 08AUG02
Class Notes and Outlines
Monday July 8

Introduction to course organization.

Student Information sheets.

Web Materials

Portfolios and grading

Some topics we will study.

Measurement and the Pythagorean Theorem [1.1]

Measuring angles, lengths and areas.

Squares, rectangles, parallelograms and triangles.

dissections, cut and paste methods of measurement.

Discuss Pythagorean Theorem (PT) and proofs.

Do Pythagorean Activity Sheet

Distribute Tangram sheets.

Assignment for Tuesday.

Read 1.1 and 1.2

1.1: 58, 11, 12, *13
Tuesday July 9

Measurement and the Pythaagorean Theorem [1.1] Review in part.

Measuring angles, lengths and areas.

Squares, rectangles, parallelograms and triangles. Circles.

dissections, cut and paste methods of measurement.

Discuss Pythagorean
Theorem (PT) and proofs.

Show video on PT

The Square Me Puzzle.

Puzzles and Polygons [1.2]

Flatland and the plane

The triangle, quadrilateral, pentagon, and hexagon.

More on measurements of angles and areas of polyons.

Activity: 1.2 Ex. 4, 5, 6

Tangrams.

Assignment for Wednesday:
Wednesday July 10
Cutting and reassembling polygons. Making puzzles.
More on Dissection Puzzles: Dissections
(Junkyard) and equidecomposable
polygons (mef)

Scissors congruence: A sc= B means figure A can be cut into pieces that
can be reassembled to form figure B.
This is also described using the word "equidecomposable".
"A and B are equdecomposable to B."

SC= is a reflexive, symmetric, and transitive relation. [like
congruence and similarity in geometry and equality in arithmetic]

Discussion: What kind of motions are used in reassembling pieces in
a puzzle?

A sc= B implies Area(A) = Area(B)

Area(A) = Area(B) implies A sc= B !!

Simple cases:

Parallelograms with common base.

Parallelograms with common parallels and same area.

Parallalograms with same area.

Triangles and parallelograms with same area.

Adding parallelograms.

Adding triangles.
Thursday July 11
Triangulation.
Film: Equidecomposable polygons.
Rigid Motions in (or about) the plane.
Orientation preserving
Translations
Rotations
Orientation reversing
Reflections
Glide reflections
Tilings Chapter 4
Regular
Activity in class: 4.1 Ex. 3, 4, 5
Semiregular
One polygonal Tile
Two tiles
Friezes
Wallpaper
Symmetry Chapter 5.
Polygonal
Planar
Tilings
Assignment for Monday:
4.1: 7, 8, 9
Monday, July 15
Tilings Chapter 4
Regular
Semiregular
Classification using vetex congruence.
How many polygons meet at a vertex?
What types can meet at a vertex?
How do the polygons fit around a triangle?
Read the "Rules" p8688
One polygonal Tile  Nonregular:
Quadrilateral Activity
Assignment for Tuesday:
4.1: 14 (based on 7)
5.1: 6 (g,h,i)
Tuesday July 16
More on Planar Tilings.
Polygonal
Card tilings
Symmetry Chapter 5.
Reflectional symmetry
Rotational Symmetry
Translation Symmetry
Glide reflection symmetry
Assignment for Wednesday:
4.1: 10, 24
5.2: 1,2
Plus symmetry of alphabet assignment.
Wednesday July 17
Modifying tilings
In class activity: Modifying tilings
Classification of Isometries
Activity: Miras for reflection one and two reflections
Video : Isometries
Every plane isometry is the product of at most three reflections.
Two reflections = rotation or translation.
Three reflections = reflection or glide reflection

Preserve
Orientation 
Reverse
Orientation 
No Fixed points 
Translation 
Glide reflection 
Fixed Point(s) 
Rotation 
Reflection 
Using Isometries to recognize symmetries of a figure or tiling.
Assignment for Wednesday: 4.2:
9,10
Bring 1 or 2 portfolio entries for review.Assignment
on symmetry.
Start on Lineland paperdue Monday.
Friezes and other patterns. (FAPP on archaeology)
Wallpaper patterns (FAPP video)
Show Video from FAPP on tilings penrose tiles
Sorry, :( I haven't written summaries for classes
from 7 18 to 722
Tuesday 723

Platonic (regular convex polyhedra) Solids

Why are there only 5?

Regular polygons around a vertex.

All verteces are "the same".

Symmetries in the plane compared to those in space an intorduction:

Translations

Rotations: Center point  central axis

Reflection : across line  across plane

Symmetries of the cube:

Rotations

reflections

rotation reflection

Side trip:

products of reflections in space:

Rotations and translations

Applications to dance

Duality

In the plane: dual tiling

Vertex  Region

edgeedge

In space: Dual polyhedron. Connected by counting and constructions.

Vertex  Face

edgeedge

Cube octahedron; Dodecahedron, icosahedron; tetrahedron.
Assignment for Wednesday July 24:
7.2: 2022
Finish 8.1: 6,9
Dual Tessellations.4.1:
24
Begin Plato essay due 729
Thursday, August 8.
FAPP Surfaces.
The hypercube.
Turning a sphere inside out.(?)
Inversion: On a line. In the plane. Orthogonal circles.
Noneuclidean universe
Escher and Hyperbolic geometry.
Not knot (?)
Flatland