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Cinderella:The Interactive Geometry Software

Jürgen Richter Gebert and Ulrich H. Kortenkamp
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Ed Sandifer
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Since The Geometer's Sketchpad revolutionized the way we teach geometry, people have seen little need for much else in the way of geometry software. Cabri has developed a bit of a following, but it and most other geometry software are usually described as "like Sketchpad, but ..." The Geometer's Sketchpad has remained the standard.

Cinderella is more. Besides all the familiar constructions of Sketchpad, Cinderella supports constructions in spherical and hyperbolic geometry, includes a theorem prover (more about that later), has more general animation features, and generates Java applets that paste easily onto web pages. You can also generate applets to create self-checking construction exercises. The creators of the software have a web site,, that demonstrates many of these features. The animation of a cycloid on a sphere is particularly spectacular. There is a nice animation of a linkage, and the self-checking construction of the bisection of a line segment gives an alluring hint of how Cinderella could be used in the classroom.

The book that accompanies the software is fairly brief, but sufficient in its instructions. It tells you how to install the software, and walks you through just a couple of examples that demonstrate the features of Cinderella. The authors seem to assume that the reader already knows something about Sketchpad, or similar software. This is a reasonable assumption. Most users will either be "graduating" from Sketchpad or learning in a classroom environment. Moreover, more online support is there for people who need to fill gaps.

The theorem proving feature of Cinderella might turn out to be a bit controversial. To quote from the documentation, "Cinderella" does not use symbolic methods to create a formal proof, but a technique called 'Randomized Theorem Checking.'" If a statement is true for very many sufficiently general examples, does that tell us it is probably true in general? Some people will be sufficiently convinced to accept the statement and move on to something else. Others may use the highly probable truth of a statement as a guide to what to try to prove. Still others will not want to trust the technique at all. The question of the validity and utility of such "proofs" could enliven quite a few faculty lounges at coffee break. It might also generate useful discussions with students.

The middle of the book spends a good number of pages explaining the mathematics behind Cinderella. The underlying data is maintained in the complex domain. Animations and constructions that exhibit discontinuities in the real domain operate smoothly in the complex domain. As a consequence, Cinderella will display both branches of certain linkage constructions, where Sketchpad displays only one.

The end of the book documents how to generate Java scripts to create interactive plug-ins for web pages. At first, the documentation may seem kind of sketchy, particularly to someone, like this reviewer, learning for the first time about .jar files and the like. Surely, it couldn't be as easy as the directions make it seem. Follow the directions carefully, though, and it works.

Cinderella makes it easy, even for Java-illiterates, to make beautiful interactive applets that demonstrate geometry theorems. The user can drag points around and see what happens to the construction when the points are moved around.

There are things that Sketchpad does better than Cinderella. Cinderella is web-friendly, and less paper-oriented, so it is harder to cut bits out of a construction to paste them into paper documents. Cinderella also isn't much good as an emergency graphics package. I often use Sketchpad for quick tasks like putting coordinate axes into a calculus quiz, and once I even used it to draw floor plans for a proposed new mathematics wing for our science building. Cinderella wouldn't do either of those very well. On the other hand, Cinderella is a versatile and powerful tool for studying and teaching Euclidean, hyperbolic and spherical geometry.

Euclid said that there is no Royal Road to Geometry. Euclid was right, but good software helps. Cinderella is very good.

Ed Sandifer ( [email protected]) is a professor of mathematics at Western Connecticut State University and an enthusiastic fan of Leonhard Euler.

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