Recently, I read again the book Turtle Geometry: The Computer as a Medium for Exploring Mathematics by Harold Abelson and Andrea diSessa. Although dated in the 1980s, it is still one of my favorite books. You can get a peek of the book content from Brian Hayes’ article. The chapter on curvature is developed from this paper of Andrea diSessa.
Those who know the graphic programming language Logo will be familiar with the turtle. You can easily get some fancy graphics on the screen by programming the turtle. If you’re a kid, you’ll be fascinated. If you’re not a kid, soon you may get bored.
However, the book shows how this turtle can lead you to explore the world of math, mostly geometry, as well as elementary number theory, logic proofs, and general problem solving skills via programming. If you follow the projects in the book, you’ll get the turtle running on 2D curve surfaces, roaming in 3D space, even venturing into the Einstein’s curved spacetime. So there can be a lot of math under the name “Turtle Geometry”.
All these wonderful activities are done by Logo-like programs, i.e. algorithms using loops and simple recursion, manipulating vectors. They are not deep, but can be quite involved, e.g. in demonstrating Einstein’s general relativity.
Message of the book is this: geometry can be studied locally, from turtle’s viewpoint. This is in contrast to the usual classroom or textbook treatment of geometry, which takes a global viewpoint. When we draw triangles, circles, or just intersecting lines on paper, we are taking the bird’s-eye view. We’ve learnt a lot of geometry from this global view, so it is refreshing to find out how much a turtle can learn from its local view. It turns out that a lot of global features can be deduced via logic if you’re a smart turtle.
Long long ago all humans were “turtles” living on the ground. Most believed that the ground is flat, but the smart one have already deduced that the Earth is round. Craftsmen made pretty accurate globes long before astronauts take the Earth-rise picture from Moon journey.
Though viewpoints may differ, the geometrical results are the same. The authors show the equivalent of these viewpoints by examples, program codes (i.e. algorithms), math calculations, even logic proofs – in words. It will be a challenge to verify this equivalence formally, via theorem-provers.
I think formalizing Turtle Geometry will be a good research project.