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.
*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.