The Archimedean solids and the semi-regular tilings of the plane are very closely related. Using geometric software and building polyhedral models and kaleidoscopes we will investigate these relationships. The participants should leave with a comprehensive understanding of the Archimedean solids as well as a full set of kaleidoscopes, the inserts for those kaleidoscopes, computer sketches, and the instructional materials to use their new understanding in the classroom.
Spherical geometry is fascinating and applicable in its own right (after all, the Earth itself is approximately a sphere). The geometry of the sphere also provides a contrast with the geometry of the Euclidean plane. Participants will explore fundamental concepts such as straightness, angles, and distance on the sphere and will create and investigate geometric figures on hands-on spheres, including the Lenart sphere. Technology will also be used as well. The course will investigate additional topics such as maps and projections, symmetry on the sphere, and the connections between polyhedra and spherical tessellations.
What would geometry be like if Euclid had neither compass nor straight edge? In this mini course we start with that question and find an equivalent construction system in the world of origami. The topics include the basic Euclidean constructions, folding regular polygons, tying knots in strips of paper, analyzing folds to understand the geometry, and building 3-dimensional models from folded paper. The activities are all hands-on, and the participant should leave with a large assortment of folded models as well as the materials needed to use the ideas in a classroom.
Transformations come to life with the use of mirrors, models and technology, including The Geometer's Sketchpad and calculators. Transformational geometry will be connected with other geometry through constructions, coordinates and problem solving. The basic properties of transformations will be investigated visually. Transformations will be used to create and analyze symmetric patterns and to analyze and simulate physical devices.