Thank you for your reflections (which I found much easier to read once I put in some tags to preserve your paragraphing. At first it was just one long breakless page!) The story of the way you came upon the source of your paintings is very non-linear in itself, and the challenge of translating the Portuguese at a late date is daunting. The presentation came out well, although as I mention below, it would have been good to identify some of the unfamiliar artists and to place things in context somewhat more clearly, something you probably would have done with some more time.

I sympathize with your ambivalence about the structure of the lectures, or lack thereof. The tone in this course was somewhat different from other years, to a large degree because of the experiment with the web base, but also because there did not seem to be a response to the problems that I introduced, at least not to the extent that I had originally thought. We never did generate the collections of exercises, along with solutions to the exercises, that I hoped for when we began. But there were enough other things developing that I did not feel that I had to enforce my preconceived idea of what we would be doing, in particular in response to the unusual intercommunication possibilities afforded by the internet. We have learned a lot, but there are many other questions that will only come after much more experimentation.

With respect to chapter 8, all I can say is that the computer doesn't seem to care whether we give it two, three, or four coordinates for every point, although it always wants us ultimately to give pictures in screen coordinates, essentially two dimensional. In three-space we can make all three axes perpendicular, but we can't in the plane, although there are some symmetrical arrangements of three lines that are useful to contemplate. Analogously we can get four lines rather symmetrically situated in three-space, for example the four long diagonals of a cube, but in four-space we could make them totally symmetrical, with a right angle between any two of them. There is just more room there. It makes you think.

Good luck with future connections between math and art.

Commentary to the Maze Group

Although your work has furthered the study of mazes and labyrinths beyond the preliminary investigation that some of you carried on in the book report exercise earlier in the semester, it does not seem that there has been that much progress in understanding the mathematical principles involved, in particular the function of dimension in the subject. It may be that there just isn't that much there, and the subject is too easily exhausted.

Let me comment on several aspects of your website. The overall structure is certainly fine, inviting the reader to explore the concept in literature, history, art, recreations, and algorithms, but there could have been more of an attempt to coordinate the form of the various sections. All of the pages seem quite wordy, unrelieved by illustrations, even though there were links to visual material.

It would have helped in the literary portion to include links to biographical information on the various authors, especially George Eliot with her Victorian connections. The Borges link is a nice one, although itself a bit rough and unscholarly in its writing. The background texture on the Auden poem makes it difficult to read, and it would have been better to analyze it somewhat, since it is a fairly good poem and it does bring up a number of interlinked images that can be related to other aspects of the topic. In particular it is one of the most "dimensional" of the literary examples in that it explores the concept of an overview. This overview notion shows up in at least one of the computer games, where a player can navigate a series of rooms without looking at the floor plan, or choose to see the plan of the maze "from above". That would have been a nice way to establish some linkages within the final project, to get away from the impression that the parts are separate somewhat unrelated entities.

The history section would have done well to link to some of the artistic representations of the Minotaur legend (there is an nice on in the "Mazes and Mathematics" link in the Recreations and Games subsection) as well as to some more serious sources of information of the encyclopedia variety. It should not be necessary to highlight the words each time they appear in the page, a very distracting feature. The link to Borges is not particularly historical, and the "Greece" link only goes to a very general source on mythology, with no further information on how to follow that thread. "Crete" appears to go to a travel folder, and "Nile" to a field trip, neither of which adds much to the narrative or gives any information that might be useful to labyrinths, mathematical or otherwise. There must be more legitimate sources out there. The material on Egypt was intriguing since mazes are not so commonly associated wht that culture. It would have been good to give the reader some more idea of places where additional material could be found to follow up this introduction.

In the art presentation, although the choice of images includes some that are quite striking, there is not enough elaboration to show how the mathematical concepts are suggested. The link to Escher for example does not lead to anything specifically related to mazes and there is little assistance provided in the text. There has to be some further biographical information provided about the unfamiliar artists, like "der Hundertwasser". The notion of a maze inside one's head is something that could be developed even further, in that the two-dimensional slice of a brain, in CAT Scan or MRI form, resembles a maze structure, which only can be navigated by moving into the third dimension, into the full range of convolutions of the brain. It is not clear to what extent the mandalas represent mazes at all. A link to the history section of the paper, with reference to the classical themes, would have held the sections together better. Hedge mazes as forms of artistic expression could receive more attention too.

The maze references in the World Wide Web could have been annotated more helpfully. Some of them are quite mathematical while others have little to add to the discussion. Singling out some of them for special consideration would have made the section more useful, especially if some of the games could have been linked to other sections of the final project.

The algorithm section, curiously enough, seemed less interactive than the earlier trio demonstrations that used hypercard stacks. Some visual illustration in the code sections would have been much more helpful in showing how to pair a simple maze up with a tree, and how to use the queues and stacks to make the various analyses. The in-class presentation made the comparison clearer, but it never had the force of the earlier demonstration. Some illustrations of weighted graphs could have motivated the Dijkstra method, and tied in better with the applications of maze algorithms to practical problems, as mentioned in the introduction.