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Mathematics on the Internet

Every year, millions of people go to science museums. We are rapidly approaching an age in which a comparable number of people have access to the Internet. The past year has seen a tremendous surge in the number of Internet users; a partial explanation for this is the popularity of so-called World-Wide Web (WWW) browsers such as NCSA Mosaic. (See the August 1994 FOCUS for more information on using NCSA Mosaic as a tool for Internet access.) You might say that the Internet has become a ``Web of Dreams'' that operates under the philosophy ``Build it, they will come.'' Using a WWW browser, it is relatively painless (in fact, it's a lot of fun!) to electronically obtain information from all over the world. The WWW browsers use ``hyperlinks'' to allow the user to pass from one site of information to another without having to worry (or even know!) about where the information is coming from.

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.

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Author: Frederick J. Wicklin <fjw@geom.umn.edu>
Comments to: webmaster@geom.umn.edu
Created: September 26, 1994 --- Last modified: Jul 21 1996