Arabesques and Geometry is part of Springer VideoMATH, a series of mathematical videos. The series, edited by T. Apostol, J.-P. Bourguignon, M. Emmer, H.-C. Hege, K. Polthier, is described as consisting of "videos covering topics from mathematics and its related fields, such as computational science, scientific visualization and mathematical physics. The level ranges from research to teaching, and some of the videos address not only the experts but all who have an interest in science."
The primary purpose of this video is to explore algebraic group structures that arise when describing certain patterns. The patterns discussed arise naturally in Arab culture and are expressed in the art form known as the arabesque. The setting of the video is the Alhambra, in Spain (a natural choice for two Spanish authors), in which almost all types of arabesques can be found. The scenes are very picturesque and provide a concrete image to associate with the mathematics being discussed. The narrator has a very soothing (almost meditative, but not monotone) voice and moves at a reasonable pace.
There are really two main parts to this video, dictated by the difficulty of the content. The first ten minutes (of twenty) are devoted to a quick, yet thorough, discussion of rigid transformations in the plane (e.g., rotation, reflection, and translation). This portion is a truly excellent motivational lesson which can be especially useful in a geometry course designed for elementary education majors. The examples and animation sequences beautifully illustrate the mathematics and would be appreciated by undergraduates. The second half, however, is a bit more advanced. It focuses primarily on the classification of the 17 plane crystallographic groups and the connection to arabesques. This portion of the presentation is perhaps more suitable for a student of abstract algebra.
In comparison to Touching Soap Films, also in the VideoMATH series, this one is better constructed. The use of technology is effective and not overdone. It is half the length and conveys the same amount of mathematics in an understandable manner. The music is not excessive, but does become rather boring near the end. A booklet summarizing the material presented in the video is also included.
Mark McKibben (firstname.lastname@example.org) is assistant professor of mathematics at Goucher College in Baltimore, Maryland. His research areas are nonlinear analysis, abstract evolution equations, and integral equations. His most recent work deals with abstract nonlinear nonlocal Cauchy problems in Banach spaces and integral equations governed by causal operators. He is the co-author of the book Algebra (with Dave Keck and Shane Rosanbalm, 1998, 2nd edition) and is currently co-authoring the texts "Real Analysis" and "An Introduction to Higher Mathematics."