MESH is the latest entry in the Springer VideoMATH series: previous titles include The Millennium Meeting Collection (2002) and Touching Soap Films (1999). As the title promises, MESH focuses on the subject of discrete geometry, particularly discrete meshes, and their theoretical and practical importance in astronomy, engineering and computer graphics.
There's an historical story line to MESH as well, which runs something like this. Discrete geometry was known to the Ancient Greeks and became a highly developed field of study which could solve describe many types of objects. However, common problems such as calculating the volume of a curved object could only be approximated using discrete measurement, and calculus was developed to deal with such situations. Calculus was once thought to have superceded discrete mathematics, but today the two approaches exist side-by-side, each with particular areas of application, and the advent of computers has given particular impetus to the use of discrete mathematics, for instance in the use of discrete meshes to model shapes.
The mathematics presented in MESH ranges from the elementary to the advanced: anyone who has completed primary school should be familiar with the Pythagorean Theorem, but Penta surfaces (which can currently be described only through discrete meshes) are probably outside the experience of most. Historical information is integrated with the mathematical presentation: for instance, the concept of Platonic Solids is introduced with the example of a dodecahedron carved before 500 BC. Other historical topics covered in MESH include the Pythagorean School, Plato's theories regarding regular solids as the building blocks of all matter, and Kepler's efforts to describe planetary motion.
The most immediately obvious quality of MESH is its astonishing beauty: independent of the mathematical content, it can be enjoyed for its visual appeal and wit. MESH is entirely computer-generated and has won numerous awards, including Best Animation at the 2005 New York International Independent Film and Video Festival, Best Experimental Film at SCINEMA 2006 (the 6th International Festival of Science Film in Sydney, Australia), and Best Scientific Visualization at the 2006 Red Stick International Animation Festival in Baton Rouge, Louisiana.
MESH will appeal to a variety of audiences on many different levels. People with a casual interest in mathematics, history or animation can enjoy it without needing to understand all the mathematics presented. It would be a good choice to show middle or high school students contemplating the pursuit of mathematics as a career. University mathematics majors will benefit on a different level, as they will understand the more complex ideas presented and may take the opportunity to follow up on some of the references listed in the accompanying booklet.
Beau Janzen holds degrees in Graphic Design and Instructional Systems Design and is a faculty member at the Art Institute of California in Los Angeles, where he teaches mathematics and computer animation. He has created animations for many types of projects, including feature films, amusement park rides, television commercials, and educational programming, including the WGBH series Evolution. Konrad Polthier is chair of Mathematical Geometry Processing at the Freie Universität Berlin and scientist and author or editor of numerous books and articles on mathematical visualization.
Sarah Boslaugh (firstname.lastname@example.org) is a Performance Review Analyst for BJC HealthCare and an Adjunct Instructor in the Washington University School of Medicine, both in St. Louis, MO. Her books include An Intermediate Guide to SPSS Programming: Using Syntax for Data Management (Sage, 2004), Secondary Data Sources for Public Health: A Practical Guide (Cambridge, 2007), and Statistics in a Nutshell (O'Reilly, forthcoming), and she is Editor-in-Chief of The Encyclopedia of Epidemiology (Sage, forthcoming).