Turtle Geometry

“… our real aim is to establish a notation for the range of complicated things a turtle can do in terms of the simplest things it knows.

… whereas we normally measure geometric figures by their interior angles, turtle turning corresponds to the exterior angle at the vertex. So if we want to draw a triangle we should have the turtle turn 120°. You might practice “playing turtle” on a few geometric figures until it becomes natural for you to think of measuring a vertex by how much the turtle must turn in drawing the vertex, rather than by the usual interior angle. Turtle angle has many advantages over interior angle, as you will see. (…)

One major difference between turtle geometry and coordinate geometry rests on the notion of the intrinsic properties of geometric figures. An intrinsic property is one which depends only on the figure in question, not on the figure’s relation to a frame of reference. The fact that a rectangle has four equal angles is intrinsic to the rectangle. But the fact that a particular rectangle has two vertical sides is extrinsic, for an external reference frame is required to determine which direction is “vertical.” Turtles prefer intrinsic descriptions of figures. (…)

The turtle representation of a circle is not only more intrinsic than the Cartesian coordinate description. It is also more local; that is, it deals with geometry a little piece at a time. The turtle can forget about the rest of the plane when drawing a circle and deal only with the small part of the plane that surrounds its current position.

A final important difference between turtle geometry and coordinate geometry is that turtle geometry characteristically describes geometric objects in terms of procedures rather than in terms of equations. In formulating turtle-geometric descriptions we have access to an entire range of procedural mechanisms (such as iteration) that are hard to capture in the traditional algebraic formalism. Moreover, the procedural descriptions used in turtle geometry are readily modified in many ways. This makes turtle geometry a fruitful arena for mathematical exploration. Let’s enter that arena now.”

Harold Abelson and Andrea DiSessa