Martin Flashman's Courses Math 103 Summer, 2003
07JUL03 to 07AUG03
Class Notes and Outlines [Work in Progress 7/29]
Monday July 7

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.

Distribute Tangram sheets.

Assignment for Tuesday.

Read 1.1 and 1.2

1.1: 58, 11, 12, *13
Tuesday July 8

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

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

Miscellaneous:

The film lists as a guide to the course topics.

The color problem.

The sphere and the Torus.

Flatland as social commentary/satire.

Assignment for Wednesday:
Wednesday July 9
Tangrams.
Cutting and reassembling polygons.
The Square Me Puzzle.
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 10
Triangulation.
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
Assignment for Monday:
4.1: 7, 8, 9
Monday, July 14
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:
Card tilings
Assignment for Tuesday:
4.1: 14 (based on 7)
Tuesday July 15
More on Planar Tilings.
Polygonal
Quadrilateral Activity
Symmetry Chapter 5.
Reflectional symmetry
Rotational Symmetry
Translation Symmetry
Glide reflection symmetry
Assignment for Wednesday:
4.1: 10, 24
5.1: 6 (g,h,i)
5.2: 1,2
Plus symmetry of alphabet assignment.
Wednesday July 16
Tilings video (FAPP)
Two tiles
Friezes
Wallpaper
Symmetry Chapter 5.
Symmetry Video (FAPP) in different contexts
Polygonal
Planar
Tilings
Modifying tilings
In class activity: Modifying tilings
Assignment for Thursday: 4.2:
9,10
Bring 1 or 2 portfolio entries for review.Assignment
on symmetry.
Start on Lineland paperdue Monday.
Thursday 717
More on Symmetries. products of symmetries
Symmetry groups of of the equilateral triangle and the square.
Isometries
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.
Monday 721
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.
Symmetry recognition Activity
Begin discussion of Space with symmetry/isometries
Tuesday 722

Symmetry in tilings activity.

Platonic (regular convex polyhedra) Solids

Why are there only 5?

Regular polygons around a vertex.

All vertices 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
Assignment for Wednesday July 23:
7.2: 2022
Finish 8.1: 6,9
Wednesday July
23
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.
Getting familiar with the octahedron/ shadows activity.
Assignment for Thursday July 24:
Dual Tessellations.4.1:
24
Thursday, July
24th:
Counting on the regular polyhedra activity.
A look at the semiregular polyhedra.
Drawing flattened polyhedra.
Cross sections of the torus. Creating a torus from two and one pieces.
Monday, July
28th:
Similarity.[ FAPP Similarity Video ]
Similarity and Editing font size applications.
Begin: Seeing the infinite... small and large.
The fourth dimension. Connection to coordinates.
Tower of Hanoi Activity
Assignment for Tuesday, July 29th.:
7/29 See assignment on 4th dimension.
6.2: 14
Make two tori: one from two annuli, one from a single "rectangle."
Tuesday, July
29th
Surfaces and "symbols": More toruses with one or more holes.
More on the fourth dimension: how does time fit into the thinking.
Assignment for Wednesday, July 30th:
See assignment on the torus and maps and surfaces.
See assignment on Zeno.
Wednesday, July
30th
Orientation in space.... The mobius band.
Torus mapping activity.
Spheres with handles. The Klein bottle?
Chromadepth
Assignment for Thursday, July 31st:
Symmetry Day.
See assignment on Projection (accepted on Thursday)
13.4 : 26
Thursday, July
31st
Other surfaces: The Klein Bottle and spheres with handles.
The hypercube (video)
Begin Projective geometry. the affine line and plane: Perspective drawing.
Assignment for Monday, August 4
10.1: 4,5
13.3: 1,2
Monday,
August 4th
More on perspective and projective geometry.
Desargues' Theorem
Euler's Formula for the plane. Accounting in Flatland.
Conics Started.
Assignment for Tuesday, August 5
13.4: 11,12,14
Tuesday,
August 5th
More on Conics.
Conics activities.
What is Dimension: Space Filling curves
Wednesday, August 6th
More on Desargues' Theorem and it's proof for the plane.
Finite
Geometry : Models and How Axioms help understand what we see.
The projective plane as a surface
Axioms and projective geometry: Duality
Thursday, August 7th.
Pascal's Theorem
Projective generation of conics. (video)
The Five Color Theorem
Classification of Surfaces
Euler's Characteristic for a surface.
FAPP Surfaces?
Introduction to inversion?
Inversion: On a line. In the plane. Orthogonal circles.
Noneuclidean universe
Escher and Hyperbolic geometry.
Flatland?