The emergence of computer graphics has sparked a renewed interest in visual mathematics in the form of films and videotapes. Many of these efforts have made use of interactive graphics.

For the past 15 years, the yearly focal point of the computer graphics world has been Siggraph, the annual computer graphics meeting of the Association for Computing Machinery. The new, refereed, computer animations shown in both the Animation Screening Room and in the evening Electronic Theater, stringently refereed technical paper and panels sessions, and huge exhibit floor now draw in excess of 30,000 people. Many of the animations mentioned below have been shown at Siggraph, which showcases animations from the science, art, and entertainment communities.

Nelson Max was one of the pioneers of mathematical visualization
movies with his ground-breaking Topology Films project in the early
1970's, when computer graphics was first used to make mathematical
movies despite the primitive state of hardware and software. His films
on curves, *Regular Homotopies in the Plane, Parts 1 and 2*, were
made using a computer-controlled oscilloscope to plot individual
points and multiple exposures to create the image of a dynamically
moving curve. His classic film *Turning A Sphere Inside Out*
[8] captures Bernard Morin's concrete description of Smale's
surprising but abstract theorem that a sphere can be turned inside out
(``everted'') in 3-space
(see 3.3)- Max created this epic of mathematical
visualization on many computers at the nodes of the far-flung ARPANET.
Later, he and Banchoff used their common visual insight to
prove that every sphere eversion has a quadruple point.

Other early efforts in this area that received wide attention were Thomas Banchoff's interactive, real-time geometrical visualization studio at Brown University. In the late 1970's and early 1980's, he and his associates produced computer animated films of 4-dimensional objects such as the award winning ``Hypercube,'' ``The Veronese Surface,'' and wireframe versions of his recent video animation, ``The Hypersphere: Foliations and Projections.'' Several of these animations were shown at early Siggraph conferences.

At Indiana University, a scientific visualization effort led by Andrew Hanson has focused on finding new ways to represent and visualize Riemann surfaces, on rendering techniques using 4D light, and on the development of corresponding interactive methods (see 3.1); this group has produced a variety of short animations, including three shown at Siggraph, ``Visualizing Fermat's Last Theorem,'' ``FourSight,'' and ``knot,'' and three others at IEEE Visualization conferences-

George Francis of the University of Illinois at Urbana-Champaign has
worked on a number of mathematical animations. Francis, Donna Cox and
Ray Idaszak of the National Center for Supercomputing Applications
created the animation *The Etruscan Venus*, shown at Siggraph 88.
A variety of other short video productions have been created
by Francis and illiView teams as well. Francis recently teamed
with Louis Kauffman of the University of Illinois at Chicago
Mathematics Department and Dan Sandin of
The Electronic Visualization Laboratory at
the University of Illinois at Chicago to produce
*Air on the Dirac Strings*, shown at Siggraph 93.

Finally, the Geometry Center has produced a wide range of animations.
At one end of the spectrum are short videos intended for use in
lieu of a computer demonstration in a lecture. In the middle of the
spectrum are longer animations by individuals and small groups. Examples
of such productions are *Twisting and Turning
in Four Dimensions* by Dennis Roseman, who has made several animations on
3D and 4D knots shown at mathematical conferences, and
*Computing Soap Films and Crystals*, distributed by the American
Mathematical Society, produced by the ``Minimal
Surfaces Team''
of pure and applied mathematicians led by
Jean Taylor of Rutgers and Fred Almgren of
Princeton.
At the other extreme are high-quality
productions aimed at the general public, which require years of
effort. These include *Not Knot,* a guided tour of hyperbolic
geometry (see 3.1)
and knot theory directed by Charlie Gunn
and Delle Maxwell (see Figure 9), and *Outside
In*, which focuses on the sphere eversion problem (see
3.3 and Figure
16), directed by Silvio Levy,
Delle Maxwell, and Tamara Munzner.

To sum up, there is today a lively activity in the area of producing mathematically oriented computer graphics animations that are shown at computer graphics conferences and mathematics seminars, and to the general public.

Thu Sep 21 19:17:33 CDT 1995