Instructors: Will Webber
Have you ever had a student ask "What is the fourth dimension?" Did you know what to say? This mini-course is designed to be a first step in answering that question. By looking at how we represent 3 dimensions on a flat (2D) piece of paper we will find ways of representing the 4th dimension in space. Topics will include projections, cross sections, and basic shapes like spheres, regular polyhedra, and uniform polyhedra. We will build as many models as we can to help visualize. We will use paper, tape, glue, string, sticks, Rubber tubing, the zome tool, and polydrons to build these models. Be ready to stretch your mind and take a peek around that corner into the 4th dimension.
How many ways can you cover a strip with squares & dominoes? What's the best way to stack oranges? The number of ways to break a polygon into triangles matches the number of certain paths through a grid -- why? Counting is a familiar and rich area of mathematics, relevant for all K-12 curricula. We will work together on projects emphasizing visual reasoning and establishing numeric patterns. Familiarity with permutations and combinations would be helpful. The course could also be called "Exploring Pascal's Triangle" -- most of what we will discuss can be found in that amazing structure.
Tessellations are where art and math meet. We'll design shapes that fit together without gaps or overlaps, squares for instance, and then turn them into interesting shapes. MC Escher will inspire us to investigate the symmetries of tilings. Artists have been weaving tessellating textile designs for millennia and potters of the Southwest pueblos have been decorating their utilitarian wares with designs that we know recognize as having great mathematical interest. The aesthetic appeal of tessellations is obvious; the mathematics is a bit more subtle. We will analyze the symmetries of repeating designs by studying the underlying transformations; translation, rotation, reflection and glide reflection. We will classify the infinity of different designs into just seventeen groups. You will even leave with a credit card sized guide to identifying all the symmetries. Geometry teachers will glean ideas to enliven their course. And no one will look at repeating patterns in quite the same way again.