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Publisher:

Springer

Publication Date:

1998

Series:

Springer VideoMATH

Price:

39.95

ISBN:

978-3-540-21385-7

Category:

Video

[Reviewed by , on ]

Patrick D. Shanahan

11/3/2000

*VideoMath Festival* is a collection of twenty award-winning mathematical videos presented at the 1998 International Congress of Mathematicians in Berlin. The videos range from one to seven minutes long and cover a wide range of mathematical topics, including classical results of Archimedes, Eratosthenes, Pythagoras, and Fibonacci, the geometry of surfaces and the universe, inversions and deformations of the sphere, random walks and ergodic theory, numerical methods applied to fluid and vehicle dynamics, and space filling curves. These topics are presented at a general scientific level and are supported by animation that ranges from comically playful to visually stunning.

*VideoMath Festival* is an eclectic mix, but one common theme of all the videos is the mathematically informative nature of the animation. This is accomplished in a number of different ways. In videos such as *Fibonacci and the Golden Mean*, *The Story of Pi*, *On Archimedes' Path*, *The Shadows of Alexandria*, and *The Theorem of Pythagoras*, ideas are presented in historical context with an emphasis on geometric verification and approximation. The mathematics is simple enough that formulas are given with justification or plausibility arguments. For example, in *The Story of Pi*, animation leads us through a simple limiting derivation the area formula for a circle by cutting the circle into n wedges, then rearranging the wedges into an approximately rectangular shape. More advanced geometric topics are discussed in the videos *Geodesics and Waves* and *The Shape of Space*. Here the emphasis is on introducing new mathematical ideas making full use of modern computer graphics. For example, in *The Shape of Space* we get to fly through a universe which is topologically a three-torus. This video gives a nice heuristic view of closed three-manifolds, reminiscent of other work done at The Geometry Center such as *Not Knot*. The computer animation is as equally stunning in the videos: *Knot Energies*, *Touching Soap Films*, *The Optiverse*, and *The Topological House*.

Of a slightly different nature are videos such as *The Law of Large Numbers*, *Vehicle Dynamics Simulation*, and *Challenges in Fluid Dynamics*. These videos provide excellent visualizations of applied mathematical systems produced by numerical methods. The emphasis here tends to be on the applications rather than the theory. For example, in Vehicle *Dynamics Simulation* we see the computer animated results of a numerical solution to a system of differential equations controlling the test drive of an yuppie-like SUV. Finally, in a category of its own is a very entertaining video called *Evolved Virtual Creatures*. In this video, we see the results of evolutions of virtual creatures grown form artificial genetic codes that have evolved interesting ways to accomplish (with varying success) different goals.

In summary, *VideoMath Festival* is wonderfully entertaining and informative collection. Most mathematical audiences will all find some part that is immensely enjoyable. There are also many possible uses for math educators. For example, videos related to numerical and graphical solutions to differential equations can be shown in class to either enhance topics covered in the classroom, to initiate ideas for student research projects, or just for pure entertainment. This video's wide range of topics, varied levels of mathematical sophistication, and stunning graphics make it a must have for any institution and a welcome addition to any personal video library.

Patrick D. Shanahan (pshanaha@popmail.lmu.edu) is Assistant Professor of Mathematics at Loyola Marymount University in Los Angeles. He researches in the field of geometric topology where his main focus is on geometric and algebraic invariants of three-manifolds.

Fibonacci and the Golden Mean.- The Story of Pi.- The Law of Large Numbers.- Vehicle Dynamics Simulation.- Challenges in Fluid Dynamics.- On Archimedes' Path.- Geodesics and Waves.- Mandelbloom.- The Animation of M.C.Escher's Belvedere.- Meditation on Homotopy of Embedding.- Knot Synergies.- Soap Bubbles.- Touching Soup Films.- The Shadows of Alexandria.- The Theorem of Phytagoras.- The Optiverse.- Homage to Hilbert.- Evolved Virtual Creatures.- The Topological House.- The Shape of Space.- Outside In.

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