jsMath: Mathematics on the Web  
While MathML certainly would be the best method of displaying mathematics in web pages, it is not available in most browsers, and so it is hard to create web pages with MathML that are viewable by a large number of people without requiring them to use a specific platform and browser. It is possible, however, to render mathematics in a crossplatform, crossbrowser manner using JavaScript, Cascading Style Sheets, and unicode characters; jsMath implements such a strategy.  
WeBWorK system components  
WeBWorK is a homeworkdelivery system designed at the University of Rochester. It is an opensource project, and at Union, we have been active in developing some significant additions to the project, including a new mathematical parser that is extensible and customizable, new answe checkers for a wide variety of answer types, 3D graphics capabilities, general systemlevel improvements, and a large number of new problems.  
Computer Graphics in Mathematical Research: from ICM 1978 to ICMS 2002  
In 1978, Banchoff gave a paper at the ICM conference in Helsinki that discussed five projects related to computer graphics. These developed into themes that he revisted over and over again, and 25 years later, in 2002, he gave a followup paper at the ICMS conference in Bejing that was a retrospective on these five topics. This poster illustrates the graphics available then and now, and gives an indication of the different mathematics that are suggested when the same object is viewed in two different ways. As with so much of Banchoff's recent work, there is an associated web site that explains the mathematics behind the that appear in the poster.  
Notes on the Fourth Dimension  
These are some animations and discussions from a class on visualizing the fourth dimension. Most of the animations are to help understand the hypercube, either through projections, unfolding, or slicing.  
Para Além da Terceira Dimensão (with  
This exhibit is based on the virtual gallery, "Surfaces Beyond the Third Dimension", descibed below. That web site was translated into Portuguese, and became a travelling exhibit, visiting over nine cities in Portugal and Brazil. It includes all the original artwork, plus two new images developed specially for this show, together with 13 new movie clips and associated descriptions. Unlike the original physical exhibit, this one incorporated computers where the visitors could interact with the various objects or view the movies. There was also an associated website, and a CDROM containing the artwork and complete web site so that viewers could take the show home with them. See the paper "A virtual reconstruction of a virtual exhibit" (6.1M PDF) for more details.  
Math Awareness Month 2000 Poster (with Tom Banchoff)  
The MAM2000 poster had two components: a printed version distributed to
mathematicians, high schools and colleges around the country, and an
electronic version available over the Web. The theme of the month was
"math spans all dimensions", and the poster revolved around a central cone
image that progressed in dimension from 0dimensional points, to
1dimensional curves, to 2dimensional planes, to 3dimensional surfaces;
the suggestion is, of course, that we can continue on to higher dimensions
as well. Around the cone are pictures of people whose work relates to
dimensions. The interactive poster
includes links to information about the people shown, the various
dimensions, and the central cone itself. There are animations to help
hyou understand the central shape, and interactive areas
where you can manipulate objects in the various dimensions. There are
links to books and other web sites where you can find more information,
and there are links to online articles about dimensions.
With Tom Banchoff, I designed the poster, both for the print and electronic versions. I developed and executed the MAM2000 web site. See "Math Awareness Month 2000: an interactive experience" (4.2M PDF) for more information about the poster and its development process.  
Surfaces Beyond the Third Dimension (with Tom Banchoff)  
This is an electronic art gallery that is based on an exhibition that
took place in March of 1996 at the Providence Art Club in Providence,
Rhode Island. Although the physical exhibit is long over, it lives on as a
virtual experience on the web. The artworks all have their origins as
computer graphics of surfaces with connections to fourdimensions. In this
electronic version of the show, unlike the original, there are animations
and virtual reality files that allow the viewer a dynamic view of the
phenomina; there are also explanations of the mathematics involved and
comments from the artist to help encourage the viewer to learn more about
the underlying mathematics of the images.
I helped to produce eight of the thirteen images shown, and I developed and executed the virtual gallery based on photographs of the original display space.  
A New Polyhedral Surface  
This is an electronic research paper that describes an unexpected
polyhedral tight immersion of the real projective plane with one handle.
This polyhedral surface is unexpected since there is no smooth counterpart,
so it represents one of only a handle of examples where the smooth and
polyhedral theories differ in a significant way in low dimenions.
The paper is an example of the type of exposition that is possible using the new hypertext techniques available on the web. It was written in 1994, when the web was still quite young, and I have resisted updating it, as I want to keep it as an historical example of the work of the time.  
Communications in Visual Mathematics (with Tom Banchoff)  
This is the prototype volume for a new electronic journal in mathematics.
Unlike most current electronic journals, which are designed to get papers to you faster and
cheaper and include only a minimum of hypertext and alternatemedia
components, the CVM is intended to explore the possibilities presented
by the new hypertext technologies. The articles are not traditional,
linearflow papers, but are constructed so that the structure of the
papers corresponds more closely to the structure of the information
involved. In addition, many include animations or interactive
demonstrations that are central to the development of the ideas they
present.
The work on this journal has been hampered by the slow development of the MathML standards for mathematical markup on the web, and the even slower development of browsers that can handle the standard. The project has been stalled because of this.  
Understanding Complex Function Graphs (with Tom Banchoff)  
This is a work in progress that investigates a new method of understanding the graph of a complex function of one complex variable as a surface in (real) fourdimensional space that has been made possible by computer graphics technology. The paper develops a mechanism of navigating the various views of the surface as projections from four dimensions to three, and maps these views onto a tetrahedron, where various paths on the tetrahedron correspond to rotations of the object in fourspace. In this way, the traditional views of the real and imaginary parts of the graph are linked by a series of intermediate steps. New insight into the nature of a complex function can be gained through the study of these sequences.  
The StageTools Package  
This is a set of external modules for the
Geomview graphics
package. They are designed to make it easy to produce complicated
mathematical objects within Geomview, and then use these objects to make
animations suitable for web pages or inclass demonstrations. The package
is described more completely in my
research statement, and in the
documentation to the package itself. See also the paper,
"The StageTools
package for creating geometry for the web" (2.6M PDF).
This program has been instrumental in
generating the graphics involved in the projects listed above.


