Fall 2001 was an intense semester, both at a global level (attack against
the U.S., war in Afghanistan) and in the limited context of MAT 103
(completion of the labs page ). Despite my half
-time status, time ended up being very limited, and the prospects for new
MAT 103 creations were rather dim. But that's what students and homework
are for, I guess: Bridgette Shepard's 'clumsily' drawn design attracted
my attention at once ... and ended up reversing my decision not to produce
a fourth MAT 103 postcard!
At first, I was sure that there was
something wrong with Bridgette's design:
how could hexagons, typically associated with 60-degree patterns, help create
a 90-degree pattern? Was some 'unintensional cheating' involved, possibly
related to her eye operation? Questioning her led to admitting that she had
essentially 'copied' figure 6.118 from my book,
another Pc'4gm design created a few years ago by (MAT 103 homework and)
former student Amber Sheldon. Copying is in fact allowed for this type of
assignment, allowing bored students to get away with little effort, while
motivated students can go as far as their imagination allows -- some
democracy here! But what did Bridgette's regular hexagons -- or were they
not regular, after all? -- have to do with Amber's overlapping squares?
Calling that 'copying' was a bit of a stretch...
Being a bit slow or lazy (or both), I could not see how to 'place' the
hexagons in order to allow for 90-degree rotation. My limited computer
skills (and graphics packages) would not make for the lack of effort and
imagination, and frustration began to set in, until my next door colleague
Linda Johnson walked by ... and gratiously allowed me into the 'wonderland'
(teaching manipulatives) Room 302; boy, weren't those wooden hexagons a
relief, demonstrating the obvious: four congruent regular hexagons may
'kiss' each other around either a square or a four-point star ... and, in
addition to two obvious 90-degree rotations and one obvious reflection
mapping any two adjacent hexagons
to each other, there is also a glide
reflection parallel to the square's or star's diameter that works -- well,
there exist no less than eight additional isometries that achieve the same
result, but let's not talk about that right now... Less obviously, any four
quadruples of hexagons may in turn 'kiss' each other in a most symmetrical
manner, and so on ... until a wallpaper pattern with reflection in four
directions (p4m) is born!
Soon I realized that Bridgette's creation deserved better than oblivion.
First came a multicolored version , then a
November 2001 exam question, together
with a 'dual' (featuring squares that create a 60-degree pattern),
and, finally, a few months later (with the assistance of MAT 103
student Robert Henning and the Oswego Printing) a postcard closer to
Bridgette's original two-colored Pc'4gm design ... all celebrating the
tenuous coexistence of fourfold and sixfold rotations (also attained in
a remarkable tile that I acquired during my Thanksgiving 2001 trip to
Barcelona ). Wonderfully, Bridgette's
four-star tiling appears to be new, while its six-star dual is well
known, found for example in page 13 of J. Bourgoin's classic
"Arabic Geometrical Pattern & Design" or in page 149 of Syed Jan Abas
& Amer Shaker Salman's "Symmetries of Islamic Geometrical Patterns"!
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