Computational Diversions columns by Mike Eisenberg.
Computational Diversions: Puzzles, problems, activities, and other things to destroy what little social life any of us might have managed to cobble together. Int J Comput Math Learning (2007) 12:81–84. https://doi.org/10.1007/s10758-007-9112-4
This column will be devoted to ‘‘recreational computing’’. I’m almost tempted to end the entire column with that one sentence, in the anticipation that since everyone will have a different idea of what that means, people might be tempted to submit all sorts of strange or kooky or gorgeous ideas. My guess is that this would be a good thing.
One of the tricky aspects of a ‘‘computational diversions’’ column is determining just how much programming should, or plausibly can, be included in a particular example. On the one hand, it is certainly possible to suggest complex research-level projects that require pages and pages of procedures; but in that case, we may be out of the realm of computational diversions and into the (admittedly nearby) realm of computational obsessions. On the other hand, very simple programs with very interesting behavior seem hard to come by.
One could hardly expect a column of this sort to go for a while (in this case, three iterations) without a mention of fractals. It’s fair to say that generating fractal designs is one of the most compelling means for introducing students to experimental mathematics. My first introduction to such figures was in Abelson and diSessa’s book Turtle Geometry (Abelson and diSessa 1980), in which the authors use Logo-style turtle walks to create the Koch snowflake, Sierpinski triangle, and numerous other beautiful self-similar patterns.
Computational Diversions: Recursion – or, Better Computational Thinking Through Laughter. Int J Comput Math Learning (2008) 13:171–174. https://doi.org/10.1007/s10758-008-9135-5/
One of the predictable side effects of writing this column is that on a regular basis—at least several times a year—I am compelled to come up with a topic, a computational diversion. I would like to pretend that this process is structured, impeccably organized, planned months in advance; but as anyone who knows me could attest, things just do not work out that way. The result is that I am periodically on the lookout for some subject matter that could make for a column—and more especially, a subject that has not already been done to death in recreational math circles.
One of the favorite display shelves in every recreational programmer’s cabinet of curiosities belongs to the Julia sets. Julia sets, named after the French mathematician Gaston Julia, have a variety of equivalent mathematical definitions; they are also remarkably easy graphical entities to generate with a minimal degree of programming.
The cognitive scientist Gerd Gigerenzer—an expert in the subject of people’s judgment and decision-making abilities—once did a fascinating experiment demonstrating the ‘‘positive value of ignorance.’’ A full description can be found in (Gigerenzer et al. 1999); here, I’ll just summarize the basic idea.
A recurring theme of this column is that just a little bit of programming can go a long way—that is, by playing with relatively short, understandable programs, one can explore wondrous, and largely uncharted, intellectual landscapes. A case in point is the famous ‘‘self-forming neighborhood’’ model devised by Thomas Schelling. Schelling (who won the 2005 Nobel Prize in economics) described this model in his remarkable book Micromotives and Macrobehavior (Schelling 1978). Numerous researchers have experimented with (and extended) Schelling’s model since that time, but it seems to me that the model is so rich in possibilities that there are undoubtedly tons of new experiments just waiting to be tried.
Just about anyone, when asked, can respond to the question ‘‘What is your all-time favorite book?’’ I’m no exception–in fact, for me, there’s no contest. The book is Turtle Geometry, by Hal Abelson and Andy diSessa (Abelson and diSessa 1980). Actually, Turtle Geometry is more than just a ‘‘favorite’’ book: it actually changed my life, a story I’ve never written about.
Recently my wife brought home from the library a charming book by Phillip Done, an elementary school teacher from California, called Close Encounters of the Third-Grade Kind (Done 2009). The book is rich in anecdotes drawn from the author’s long experience in the classroom, and acts as a wonderful complement (or maybe a tonic alternative) to reading in educational research. Done doesn’t write like a researcher, but his tales are filled with the sounds and smells and emotions of the classroom, and he has–or has developed over time–tremendous insight into the personalities of the children with whom he works.
This installment of the computational diversions column introduces a new game (at least I think it’s new, and original–I haven’t seen it anywhere before). The game is called HullGrams, which is intended to suggest a blend of the classic mathematical pastime of tangrams with the geometric notion of a convex hull.
In this column, I’m giving away free ideas. There’s a certain sense of regret in doing this: these are small-scale project ideas that I’ve hoarded away for a while, hoping to see them brought to fruition. Occasionally, I’ve tried to assign one or another of these as special projects to students, but without success. Now, I’ve reached the point at which my backlog of unaccomplished ideas is so vast that it’s hard for me to worry about questions of credit, or academic glory (which is in any event sort of a contradiction in terms). So, for any reader–especially a young, energetic reader–searching for a fun and perhaps ultimately publishable project idea, please do consider the possibilities presented here. My treat.
Puzzles are an odd form of literature. For the most part, puzzle-writers—even the best of them—simply aren’t concerned with narrative, plot, character development, or even simple plausibility. Take, for example, the ‘‘island of truth-tellers and liars’’ found in so many logic puzzles: what kind of a place would really and truly be populated with (exclusive) truth-tellers and liars, anyhow? Could such people exist? (Even my best friends aren’t exclusive truth-tellers, and even my least favorite political figures aren’t exclusive liars.) And squeezed together on an island, no less—wouldn’t that lead to civil war? Or the farmer crossing the river—what’s the man doing walking around with both a fox and a goose? Maybe he should just invest in a muzzle for the fox?
Fans of ‘‘mathematical games’’ (I’m one) often end up with a small library of books on the subject, collectively containing descriptions of a vast trove of games. There are board games, or informal pencil-and-paper games, with applications to (e.g.) graph theory, number theory, combinatorics, logic, and many other branches of mathematics. In fact, the range and variety of mathematical games is so wide—and so rich in content—that one could plausibly base a pretty good introductory mathematics curriculum on these pastimes. Perhaps a few of the more advanced subjects would be underrepresented in this exercise (real analysis? differential geometry?); but a typical library of mathematical games provides, overall, a broad portrait of mathematics in general.
This essay returns to the topic of spherical turtle geometry, but in a more participatory spirit. As of this writing, we now have a Web-accessible programming interface for the spherical turtle, programmed by Antranig Basman and Michelle Redick at the University of Colorado; that’s just a formal, academic way of saying that you, the reader, can now bring up a working spherical turtle interface and write programs for it.