The Looping Theorem in 2D and 3D Turtle Geometry

In their book Turtle Geometry, Abelson and diSessa formulate and prove the POLY Closing Theorem, which gives an exact condition for when a path produced by the POLY program closes (initial and final turtle position are equal) properly (initial and final turtle heading are equal). The POLY program repeats a translation (Move command) followed by a rotation (Turn command). Their Looping Lemma states that any repeated turtle program is rotation-symmetry equivalent to a POLY program. The POLY Closing Theorem and Looping Lemma are useful in understanding and creating artistic motifs because repeating the same turtle program so that it closes properly, leads to a rotationally symmetric path. In this article, we generalize their result to 3D. A surprising corollary is that when repeating a non-closed non-proper turtle program, its path is closed if and only if it is proper.