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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