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by Dave Youngs.
Jim Wilson, a colleague here at AIMS, has always maintained that classrooms
should be full of interesting mathematical objects. Almost 20 years
ago Jim gave me a copy of a disappearing elves puzzle for my fourth-grade classroom.
Students fascination with that puzzle made me a believer. (I still have and
use that puzzle!) The puzzle presented here is my attempt to provide you with
one of those fascinating mathematical objects that is sure to captivate your
students.
This month's puzzle is a powerful visual illusion of teddy bears on
an impossible staircase. Readers familiar with the work of the late
Dutch graphic artist M. C. Escher might assume that the original idea
for this illusion came from him. His 1960 lithograph, Ascending
and Descending, which sold millions of copies and is one of his
best-known prints, uses a similar impossible staircase with monks
climbing and descending an impossible staircase circling a building. The
monks going clockwise around the staircase never reach the top of the
stairs and are locked in an infinite ascent. Likewise, the monks going
counterclockwise are locked into a less taxing, but no less monotonous,
infinite descent. While Escher did indeed make the impossible staircase
widely known to the world in his 1960 print, the original idea for the
print came from the field of recreational mathematics.
In 1958, L. C. Penrose, a geneticist, and his son, Roger*,a mathematician, co-authored an article published in
the British Journal of Psychology describing two impossible
objects. The first of these objects was the impossible tri-bar. objects.
(See Tri-bar Trauma in the Dec. 1994 issue of
AIMS for information on the tri-bar.) The second object was
the impossible staircase. This staircase was drawn in a closed loop and
had no bottom step when allowing the eye when allow in the eye to move
around it in a clockwise direction. If the eye followed the loop in an
anti-clockwise (the British term for, counterclockwise) direction, no
top step existed. This father and son team had invented two interesting
mathematical objects, just for the fun of it, and they
wanted to share them with others through their article.
Escher who was keenly interested in impossible worlds and objects
came across the Penrose article shortly after it was published. He used
both of the Penroses' mathematical creations, producing two of his most
successful print, the aforementioned Ascending and
Descending, and Waterfall, a 1961 lithograph which
cleverly incorporated three impossible tri-bars to create the illusion
of a self-sustaining waterfall.
Up and Down the Staircase is presented here to help you
build your supply of interesting mathematical objects. Each student
should receive a copy of the puzzle to take home and share with family
members and friends. An enlarged copy of the puzzle could be placed
somewhere in the classroom.
The challenge in this puzzle is to see if you can figure out how the
illusion works. I don't believe that it is important for students to
come up with the "correct answer" to this puzzle -- it happens to be
fairly complicated. I am more interested in helping students understand
that the idea for this puzzle is mathematical and that it
was invented by mathematicians just for the fun of it. In
this light, the puzzle has value simply as an affective tool to create a
more positive mathematical environment in the classroom. It has the
power to help students see that mathematics is much more than arithmetic
and that mathematics can, and should be, fun!
Readers who can't wait, can find a detailed explanation of the
impossible staircase in chapter 13, "Worlds That Cannot Exist," of Bruno
Ernst's wonderful book, The Magic Mirror of M. C. Escher.
This book has recently been reprinted for Barnes and Noble and features
Ascending an Descending on its cover. If you cannot find a
copy of this book in a local book shop, it is available via the Internet
(http://www.amazon.com). I would highly recommend that you add this book
to your personal library.
Worksheet
*Sir Roger Penrose, who invented the impossible staircase with
his father, is one of the top mathematicians and scientists living
today. He was recently knighted for his outstanding contributions in
these fields. Penrose received his Ph.D. at Cambridge in algebraic
geometry and is a professor of mathematics at Oxford University. He is
also a physicist and cosmologist and in 1988 won, along with Stephan
Hawking, the prestigious Wolf Prize for Physics. One of Penrose's
passions is recreational mathematics. This passion was stimulated from
an early age in his family by his mother, a doctor, and his father, a
medical geneticist who used mathematics for both work and
recreation.
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