Summer, 1997

## Daily Topic Schedule

 Monday Tuesday Wednesday Thursday 1-Rigid Plane Geometry 26/ 27/Introduction/ Pythagorean theorem The puzzle problem Regular polygons. 1 figure tilings. 28/Tangrams/ regular 1 figure tilings Equidecomposable Polygons Symmetry & Isometry: Introduction 29/semiregular 2 figure tilings/ 2 Rigid Space Geometry Planar and Space. Similarity. 2/Generation of Isometries/ classification 3/Isometries and symmetries- generating tilings from more interesting figures. Introduction to Space: cross sections and casting shadows. 4/ Regular and semi-regular polyhedra Symmetry in space Begin Similarity and Orthogonal Projection 5/More on regular and semi-regular polyhedra Space Isometries 3 Inversion. Projective Geometry. Topology of Planar Networks 9/More Similarity Applied Projections:Orthogonal vs. Central. Coincidences Networks in the plane (sphere). 10/Begin Inversion. More on Projective geometry- The Projective Line & Point at Infinity. The Euler Formula for the plane (sphere) 11/The Torus- flattened. Networks on the Torus. Perspectivities and Projectivities. Inversion - Circles and Lines. 12/Application of Inversion Dimension - coordinates The hypercube. The Utility Problem. 4 Topology of the plane and space. Surfaces. Non-Euclidean Geometries 16/ The Color Problem Non-Euclidean Geometry and inversion. Perspective and projection in drawing. 17/ The Projective Plane Finite Projective geometries. Duality Mathematical Models Uncountably infinitely many points on a line segment. Space Filling Curves- Fractals. Begin: the Mobius Band and the Klein Bottle 18/More on Fractals The Conics - Euclidean view. Desargues Configurations 19/Closing - Bianchon and Pascal Conic Configurations The Conics - Projective View Orientable and non-orientable surfaces. Classification of Surfaces. Turning the sphere inside out. Handout- Some details on the five color theorem.

## Overall Tentative Topic List

1. Introduction.
1. What is a Visual Argument?
2. Planar Issues
1. Points, lines and polygons.
2. Planar Tilings
3. Symmetry and Isometries.
3. Introduction to Space.
1. Polyhedra
2. Isometries
4. Magnification and Similarity
5. Projective Geometry
1. Configurations
2. Duality
6. Non-Euclidean Views
1. Projective Geometry
2. The Poincare Model
7. Curves and Surfaces
1. Continuity and Dimensions
2. Accounting for Geometry (Planar and Non-Planar)
3. Classification Problems
4. Color Problems