Turtle geometry: The computer as a medium for exploring
Turtle geometry is a book by Hal Abelson and Andrea diSessa that is a textbook on advanced concepts in explorations of mathematics more generally and geometry in particular with a strong emphasis on advanced geometry.
The book builds on the work of Seymour Papert in Mindstorms is similar to and highly influenced by the logo programming language. The book going through a series of examples and exercises to explain concepts in geometry by encouraging experimentation and exploration. The book is designed to facilitate exploration of concepts with a computer and is largely made up of programming examples -- although they are not written in any programming language that exists (and instead use a simplified, Logo-like syntax and set of examples.
The book describes turtle graphics in which a turtle represents a kind of cursor or pointer in space. Turtles have three attributes: (1) a position, (2) an orientation, and (3) a pen (which is either up or down).
The major difference between turtle geometry and coordinate geometry (which is more like the type of geometry that most people are taught) is that turtle geometry. The authors argue there are several important differences:
- Turtle geometry focuses on intrinsic versus extrinsic descriptions. In other words, turtle geometry would describe a circul as composed of repeated steps of one step forward, a turn one degree rather than an extrinsic explanation that provided the formula for a drawing a circle in terms of x and y coordinates. In other words, the quality of the figures in question depend only on the figures and not on a relationship to an outside frame of reference.
- Additionally, turtle geometry is local, rather than global in that it deals with geometry one little piece at the time.
- Finally, it is focused on procedures rather than equations with a different approach but is much better suited to the computer-based and exploratory means described by Abelson and diSessa.
The rest of the book is basically structured explorations with questions posed and hints (and sometime answered) offered. The book starts out slow with planar geometry and introduces the turtle geometry version of the closed-path theorem and its proof. It then builds a series of lemmas, explores the structure of radial symmetry and its structural visibility in turtle geometry programs, and goes through a series of proofs explained in turtle geometry terms. The book continues through examples (essentially differential geometry of various types) introducing increasingly complex concepts throughout. It proceeds through turtle geometry on spheres and others surfaces all the way to general relativity.
Theoretical and practical relevance:
The book is a tour de force, if a somewhat puzzling and unpopular one. It has only been cited 250 times, or so, since it's publication three decades ago. The audience is supposedly high school or college students but the material is both very basic and works up to dizzying complex. The terms are simple but the conceptual complexity remains on some levels (the ideas are powerful) and the result is that to grok it well, a student must still be extremely capable. The book is written in a programming language that did not (and still not) exist exactly and requires translation that offers a strange translation.
It is a brilliant and clear example of the power of Logo and constructionism more generally. It shows -- proves even -- that the simple ideas at the heart of Logo can be scaled (quickly) to dizzying complex subjects. It is a template for what a constructionist guidebook should look like. Published in the early 1980's, the book may very well have simply been ahead of its time.
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