Since the LOGO Turtle took his first step he has been mathematically confined to running around on flat surfaces. Fortunately the physically intuitive, procedurally oriented nature of the Turtle which makes him a powerful explorer in the plane is equally, if not more apparent when he is liberated to tread curved surfaces. This paper is aimed roughly at the High School level. Yet because it is built on intuition and physical action rather than formalism, it can reach such "graduate school" mathematical ideas as geodesics, Gaussian Curvature, and topological invariants as expressed in the Gauss-Bonnet Theorem.
