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A lovely book on how to think geometrically and algorithmically, using a simple programming language to produce pictures and prove theorems, starting from Eucliedean Geometry and ending with the curving space-time of Einstein's theory of General Relativity.

Sounds like a tall order! But the reader is led to develop the mathematics herself, by working through examples, learning to develop and prove theorems and write programs to explore examples. The language is LOGO, which was designed for teaching elementary school children to program. The semantics of LOGO are quite advanced: but the syntax is straightforward and, as with the mathematics, the reader is led more-or-less gently along the path of acquiring the sophisticated ideas.

A wonderful book.

(See also The Shape of Space, Visual Modeling with Logo and Exploring Language with Logo) ( )

Experience is an important ingredient in discovery. The abundance of the phenomena students can investigate on their own with the aid of computer models shows that computers can foster a style of education where "learning through discovery" becomes more than just a well-intentioned phrase. Computers can bring to learning the essential element of surprise, for despite the popular belief that a computer can never surprise its programmer since it does only what it was programmed to do, even very simple algorithms can and often do produce unexpected striking results. Encountering one of these results, studying it, and understanding how it comes about can be an open-ended adventure very different from most of the ""discovery methods"" in teaching, in which the teacher knows beforehand precisely what is supposed to be "discovered".
This book is a computer-based introduction to geometry and advanced mathematics at the high school or undergraduate level. Besides altering the form of a student's encounter with mathematics, we wish to emphasize the role of computation in changing the nature of the content that is taught under the rubric of mathematics. We wish to demonstrate curriculum that shows the computational influence in its choice of ideas as well as in its choice of activities. Some of the major themes of the book illustrate the point: representation, local-global dichotomy, linearity, state and state-change operators. Most important in this endeavour is the expression of mathematical concepts in terms of construction, process-oriented formulations, which can often be more assimilable and more in tune with intuitive modes of thought than the axiomatic-deductive formalisms in which these concepts are usually couched. As a consequence we are able to help students attain a working knowledge of concepts such as topological invariance and intrinsic curvature, which is usually reserved for much more advanced courses.