Capturing congruence with a turtle

This note compares turtle geometry and Euclidean geometry with respect to their treatment of similarity and difference of plane figures. It is observed that while the Euclidean notion of congruence faithfully captures a common perception of sameness, the turtle expression of this idea is too weak. To deal with this insufficiency we add a new turtle operation, FLIP, which turns the turtle upside down. This brings the turtle's power to express invariance of shape up to Euclid's.The problem and its solution are viewed briefly from the perspectives of mathematics, computer science and education. The mathematical discussion compares the turtle group and the Euclidean group. The computational discussion focuses on the issue of expressive power of a language and how it may be enhanced. The educational discussion suggests a classroom implementation of the above ideas.