# Martin Flashman's Courses- Math 103 Summer, 2002 08-JUL-02 to 08-AUG-02 Class Notes and Outlines

 7/8 7/9 7/10 7/11 7/15 7/16 7/17 7/18 7/22 7/23 7/24 7/25 7/29 7/30 7/31 8/1 8/5 8/6 8/7 8/8
Monday July 8
• 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.
• Do Pythagorean Activity Sheet
• Distribute Tangram sheets.
• Assignment for Tuesday.
• 1.1:  5-8, 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.
• Do Tangram Activity.
• Assignment for Wednesday:
• 1.2: 1-3, 8, 9

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.

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?
• One polygonal Tile - Non-regular:
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 paper-due 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 7-22

Tuesday 7-23
• 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
• edge-edge
• In space: Dual polyhedron. Connected by counting and constructions.
• Vertex - Face
• edge-edge
• Cube- octahedron; Dodecahedron, icosahedron; tetrahedron.
Assignment for Wednesday July 24:
7.2: 20-22
Finish 8.1: 6,9
Dual Tessellations.4.1: 24
Begin Plato essay due 7-29

Thursday, August 8.

FAPP Surfaces.
The hypercube.
Turning a sphere inside out.(?)
Inversion: On a line. In the plane. Orthogonal circles.
Non-euclidean universe
Escher and Hyperbolic geometry.
Not knot (?)
Flatland