We at the Geometry Center are using the Web as part of our effort to make mathematics more accessible and understandable to non-mathematicians. Documents are written in hypertext, which allows users to ``click'' on certain words, phrases, and pictures in order to obtain additional information. Our hypertext documents are designed to present mathematics to the masses using a combination of expository prose, computer-generated graphics, and animations. Furthermore, we have used the unique capabilities of the Web to produce interactive documents that allow users to explore and discover mathematics for themselves. We have a very popular Gallery of of Interactive On-Line Geometry in which Internet explorers investigate and visualize the mathematics of quasiperiodic tilings of the plane, wallpaper groups, geodesics on manifolds, Riemann surfaces, and symmetries of the hyperbolic plane. The Geometry Center's WWW documents are accessible from our so-called ``home page.'' In fact, this article is available on-line as well as part of the Geometry Center's Online Multimedia Document Library!

The Geometry Center is also looking at how to use the World-Wide Web
in order to disseminate research-level mathematics to professional
mathematicians. Already we live in an age in which a large portion of
research mathematics is distributed in preprint form, thanks to
for formatting equations, electronic drawing programs for producing
figures, and laser printers for putting it all on paper. Color
printers and copiers now allow mathematicians to include color figures
in preprints, thus increasing our ability to convey information about
geometric structures. Yet our preprints and journal articles are
fundamentally limited to static images. No paper-based preprint or
journal allows authors to publish an animation showing a rotating
surface. None allow the reader to interactively alter parameters in a
model and to subsequently generate a *new* figure that perhaps
even the author has never seen. However, these abilities are available
in hypertext documents, and the Geometry Center has begun to
experiment with these possibilities through its Web documents.

As an example, we highlight a recent mathematical result by Center postdoc Davide Cervone. Cervone discovered a certain immersion of the real projective plane with one handle into three-space that settles a long-standing conjecture in simplicial geometry. The result is being submitted to a traditional journal, but because the fundamental result concerns exhibiting a surface with special properties, Cervone chose to write a preprint using hypertext as his medium. Cervone's document begins with a highly accessible introduction to the problem (complete with hyperlinks to basic definitions!), and then proceeds to present and prove his new result. He then provides the reader with multiple ways of visualizing and interacting with this new immersion; there are movies of the rotating surface and of a family of level sets, and there are interactive pictures that the reader may manipulate in order to see the surface from a desired viewpoint. We are currently considering how to use this powerful medium to communicate mathematics in curricular material for undergraduates in mathematics, science, and engineering.

We believe that our presentation of mathematics on the Web is successfully reaching thousands of people from all walks of life, but it is very difficult to quantitatively assess the impact of our efforts. We have records of hundreds of thousands of connections to the Geometry Center WWW documents, but we cannot evaluate who these people are, nor determine their backgrounds. Are we reaching teachers from high schools and community colleges? Are we reaching populations that are underrepresented in mathematics? Are we making a difference in the way that ordinary people think about mathematics? Are we affecting the way that mathematicians think about presenting mathematics? We have some (very positive) comments sent to us by users, but it is difficult to extrapolate from this data. We need to understand how to better assess our outreach through the Internet.

Author: Frederick J. Wicklin <fjw@geom.umn.edu>

Comments to:
webmaster@geom.umn.edu

Created: September 26, 1994 ---
Last modified: Jul 21 1996