The "vision test" was designed for the
first hour exam in MAT 103 (Spring 1997) as a "p1" wallpaper pattern, that
is, a two-dimensional pattern having no symmetries other than translations.
Out of about 40 students that took the exam, only one did agree with my
classification; there were many other answers, relying on non-existent 90
or 180 degree rotations, vertical or horizontal glide reflections, etc.
The "vision test" turned into a post-card produced at James' Printing of
Oswego in August 1997; it is, incidentally, my first computer-drawn
pattern. One of several friends and colleagues who received the post-card
was William Huff, an acquaintance from the Art and Mathematics 97 conference
at SUNY Albany, professor of Architecture at SUNY Buffalo; his work is
mentioned, among other places, in Douglas R. Hofstadter's "Metamagical
Themas: Questing for the Essence of Mind and Pattern" (Basic Books, 1985).
Professor Huff did in turn send the post-card to a former student, Dr. P.
M. Catipovic, who wrote her dissertation on the 17 plane crystallographic
groups (the "wallpaper patterns" mentioned above).
To my ultimate humiliation and delight, Dr. Catipovic reclassified the
"vision test" as a "pg" pattern: indeed there exists a previously unseen
diagonal glide reflection, its axis running either between Z's and reverse
N's or between N's and reverse Z's! As Dr. Catipovic pointed out, it helps
to hold the post-card diagonally,
the Z's "spine" looks vertical.
Another way of arriving at the same result has been subsequently pointed
out to me by my colleague Douglas Mennella, a visiting assistant professor
who received his Ph.D. in Algebraic Topology from SUNY Binghamton;
observe that every 4x4 square with a "Z" diagonal moves to an identical
square via reflection about that diagonal and subsequent vertical
translation by one unit: the combined effect (composition) of the diagonal
reflection and the vertical translation is the diagonal glide reflection
noticed by Dr. Catipovic.
(I am indebted to Dr. Catipovic and Prof. Mennella for their alternative ways
of looking at the "vision test"; Doug has also helped me to incorporate it
into my page long before I became familiar with the needed software.)
The "Vision Test" in full