# Martin Flashman's Courses- Math 103 Summer, 2003 07-JUL-03 to 07-AUG-03 Class Notes and Outlines [Work in Progress 7/29]

 7/7 7/8 7/9 7/10 7/14 7/15 7/16 7/17 7/21 7/22 7/23 7/24 7/28 7/29 7/30 7/31 8/4 8/5 8/6 8/7
Monday July 7
• Introduction to course organization.
• Student Information sheets.
• Web Materials
• 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.
• 1.1:  5-8, 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:
• 1.2: 1-3, 8, 9

Wednesday July 9
• Tangrams.
• Do Tangram Activity
• 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.

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?
• One polygonal Tile - Non-regular:
• Card tilings
Assignment for Tuesday:
4.1: 14 (based on 7)

• Tuesday July 15
• More on Planar Tilings.
• Polygonal
• 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 paper-due Monday.

• Thursday 7-17
• 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 7-21
• 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 7-22
• 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: 20-22
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
• edge-edge
• In space: Dual polyhedron. Connected by counting and constructions.
• Vertex - Face
• edge-edge

• 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 semi-regular 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?

Assignment for Thursday, July 31st:
Symmetry Day.
See assignment on Projection (accepted on Thursday)
13.4 : 2-6

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.
Non-euclidean universe
Escher and Hyperbolic geometry.
Flatland?