Bicycle Tracks Which way did the bicycle go? Geometry in action This problem comes from the course Geometry and the Imagination taught at Princeton by John Conway and William Thurston. It emphasizes a number of levels of visual reasoning. First the identification of which track is the back wheel and which is the front. This is intimately related to basic experiences of how ‘turning’ (not straight) works. For students, I connect it with what they see with a turning bus or transport which makes a wide sweep with the front wheels and much shorter turn with the back wheels. In essence, the back wheel goes straighter than the front wheel. The only time the two tires follow the identical track is when they are both going straight. The second step is to realize that the wheels are tangent to the path at any point on the path. In particular, for each point of the track of the back wheel we can find the direction of the cross bar (which is linked to the direction of the back wheel). The final step is to realize that several points would require different lengths for the cross bar of the bicycle. I use this exercise in my geometry course after several hours of exploration of ‘what is a straight line’. The same analysis would work on other surfaces such as the sphere. Which way did the bicycle go?

Geometry and Symmetry Class 1

A geometer sees … and dreams I warn my students that if they spend too long around me, I will try to make some of their wildest dreams into dreams involving geometry. It is true that overtime, mathematics can infiltrate into ones dreams. One can problem solve (or attempt problem solving) in ones dreams. Some of my wildest dreams are about geometry!

Geometry with ... symmetry Hang a weight on a string What do you expect to see? Why? Why is a point in the middle not moving? Balance of forces: symmetry Pierre Curé’s Law: The symmetry of the input matches the symmetry of the output.

Geometry with symmetry Some dynamic images Use dynamic geometry in my own work Geometer’s SketchPad (GSP) Mix visual and kinesthetic: Exploring ‘equal lengths’, and ‘folding’ in GSP GSP Exploration ‘Equal length’ is about transformation to superimpose end points!

Geometry with symmetry Symmetry as the core of geometry. Ways of testing a proposed straight line: Carpenter’s test: Gaps means the two boards are curved. Does no gaps mean the two boards are straight?

the practice of Mathematics Goemetry not proof! Quotes on Proof and the practice of Mathematics Axioms and proofs differ from mathematics as medicine differs from food. Paraphrase of Gian-Carlo Rota, 1997 Only professional mathematics learn anything from proofs. Other people learn from explanations. Raol Boas, 1980 Any fool can know. The point is to understand. Albert Einstein