Vision

Doug Hofstadter stated a vision like the one below for visualized group theory in general. I adopt it as a vision for the Group Explorer project, and hope I restate his ideas faithfully.

For many years following Newton's and Leibniz's inventions of the calculus, the results of their work were not nearly as common knowledge as they are today. Their work was read by the very educated, but today calculus is taught in nearly every high school. This transition took place not because high school students today are smarter than the intellectual elite of the 19^th century, but rather because the exposition of calculus has improved tremendously. Similarly, the theory of relativity was once quite esoteric, but now every undergraduate physics major studies it.

Since first appearing in Evariste Galois's work on the unsolvability of polynomials by radicals, group theory has been polished and refined. Now nearly every undergraduate mathematics major is exposed to group theory. Group Explorer seeks to push the exposition of group theory yet one step further, making group theory readily accessible to a wider audience.

We are not alone in this goal; the MAA's New Mathematical Library contains a book called Groups and Their Graphs, which also seeks to make group theory visual using tools like Cayley diagrams. In the preface, the editors write

This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and understandable to a large audience of high school students and laymen.

Is group theory necessarily harder than calculus or relativity? Perhaps not, but we cannot see this readily because the exposition of group theory is not sufficiently clear. Group Explorer seeks to be an important step in improving that exposition.

Goals

Here follows a list of where the Group Explorer project is going in the coming months and years. (On a related note, but in the other direction, the History page chronicles the development of the project to date.) Most of the improvements below are scheduled to begin in the fall of 2004, and continue at whatever pace I am able to implement them.

Major enhancements

Rebuild the entire program in a cross-platform fashion, so that Macintosh and *nix versions are readily available, and behave just as the Windows version does. Release it open-source.

Load all groups at startup, enabling several functionality enhancements, including

automating the recognition of smaller groups in larger ones and the recognition of quotient groups,

drawing diagrams of homomorphisms between Cayley diagrams of two different groups,

and browsing the group library within the program itself, searching/sorting by properties of groups, etc.

Redesign the internal representation of a group to allow for arbitrary finite groups limited only by machine size (i.e., not by their embedding in S₂₀). This will facilitate more efficient group operations, a more flexible .gp file syntax, and a richer Group Library.

Cayley diagrams that can be stretched and altered by clicking and dragging their parts, then saved as custom diagrams. Also, an interface for defining a group by creating a Cayley diagram, and having Group Explorer write the group definition and .gp file.

Enable Group Explorer to give the user topical advice about what they're investigating, either when asked for it or when the user seems to be consistently investigating a certain topic.

Minor enhancements

Allow more sophisticated names of groups and elements (e.g. r²s^-1 instead of rrrssss).

Remove some of the limits on custom Cayley diagrams and objects of symmetry.

Improve my graphics routines to take better advantage of graphics hardware.

Create an email distribution list for those interested in Group Explorer to join, be alerted of new developments, etc.

And many others