In the Fall of 1997, art professor Pippa Drew of Dartmouth College visited
SUNY Oswego for a general audience hour lecture and a MAT 103 lab that
she and I developed and ran together. The lab was centered around the
creation of tilings by students working in small groups and using colored
triangles; such paper triangles were routinely used by Pippa Drew and
mathematics professor Dorothy Wallace (also of Dartmouth College) in their
"Pattern" interdisciplinary course.
During the 11/25/97 "pre-Thanksgiving" lab, we used 60-60-60 triangles
(appropriate for tilings having 60 or 120 degrees rotation) and 90-45-45
triangles (appropriate for tilings having 90 or 180 degrees rotation).
There was no time for the 90-60-30 triangles that professor Drew had also
brought along, so I decided to use them on the last day of classes of the
following semester (5/5/98): I was rather excited about the experiment,
for such triangles allow for both kinds of tilings, depending on
how the triangles are "glued" to each other.
My expectations were not unjustified: several beautiful tilings of "both
kinds" (180 degrees and 60 degrees) were being created by several small
groups of students working together. I kept walking among the various
groups, encouraging them to produce more and more tilings and hawkishly
keeping track of everybody's progress. But there was one tiling, produced
by a "front row" group consisting of Erin McGivney (communications major)
Arielle Muth (art major) and James Rood (accounting major), that
attracted my attention: it looked different from anything I had seen
before, and I decided not to let it fade from my memory after the colored
triangles would inevitably go back into their envelopes at the end of the
So I drew the targeted tiling shortly after class on MacDraw, a rather
archaic drawing package that I had not yet outgrown -- a 1991 pun by my
MacDraw tutor and former Calculus student Scott Cole! -- and naturally
threw it with no hesitation into the final exam: as her question during
the test indicated, at least one of its creators failed to recognize it,
even though, in the end, she correctly classified it as a pm'g.
My student's failure to recognize the tiling made sense for at least one
good reason: due to MacDraw's (and my own) technical limitations, the
triangles employed in my drawing were no longer 90-60-30 triangles, but
closer to 110-45-25 triangles... But, in addition to possibly making an
exam question a bit more challenging for one student, my failure to
reproduce the tiling with 100% accuracy had another positive,
the "waves" created by the consecutive oblique sides of opposite-facing
triangles (and the vertical glide reflections) look smoother and
faster in the "defective" design than the ones in the "original" !
How does one measure such "fluency" was, and still is, unclear to me, but
I liked the "motion" created enough to decide that the time for a second
MAT 103 postcard had come: SUNY Oswego art professor David Fox, whom I
still have to meet in person, helped me to change black and white --
remember, I was still in my MacDraw days -- into dark blue and light blue
(corresponding to water and sky, if you wish) over the web, and "Waves"
finally found its way into "colorful" Oswego Printing in early February
1999; no, the colors on the postcard are not the ones used by the three
students (which I no longer remember).
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