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This is How We Roll
The Square Wheeled Bicycle in Theory and Practice 
The endeavor
As part of the MATH 489 Special Topics: Mathematical Modeling course at St. Norbert College in Fall 2007, twelve students collectively built a square wheel bicycle and rode it. Smoothly!
Some history
An early reference of the mathematics behind rolling a square wheel smoothly was found in an article titled "Rockers and Rollers" by G.B. Robinson published in Mathematics Magazine in 1960. In 1992, Stan Wagon, a mathematics professor at Macalaster College in St. Paul, Minnesota, published another article on the topic in the same journal. In 1997, Wagon built and successfully rode a square wheel tricycle. This tricycle was pedaled over a road of catenary curves. In 2004, Ken Moffett refurbished Stan Wagon's tricycle smoothing out some of the engineering aspects.
Our challenge: Change the road
At first, our class thought the bicycle itself needed to be altered in order to make the square wheels roll. Some considerations included different wheel sizes or angling the bicycle. In order to think beyond the bike, our class observed the basics of a bicycle by rolling a Spider Man bicycle, owned by our instructor's 5-year old son, across a table. We observed that the center of the bike's wheel was always parallel to the table. In addition, the wheel made contact with the table at only one point on its surface as it rolled. Yet, a square wheel would make contact with a plane surface at each point along its side. An "ah-ha" provided that it was not the bike that needed to be changed but the road!
Bike construction 
We found the most challenging engineering component of creating a square wheel bicycle to be the design of the wheels. They had to be sturdy, yet not too heavy, and roll with the turning of the pedals. We began with a traditional bicycle. To utiltize the existing mechanics, we decided to reuse the bike's axles. Also, the diagonal of our square wheel had to have the same dimension of the diameter of the round wheel to fit within the fenders. We measured one of the radii of the round wheel which became half the diagonal of the square. Then, using the Pythagorean Theorem, the sides of the square were calculated. Finally, wood was cut using these dimensions. The back wheel consisted of two wood squares which were connected with four small triangles in each corner to give the wheel more width, making it more stable. To aid the placement of these small triangles, the two squares were cut in half to ease the assembly. The axle crossed the middle of the wheel and was hammered to the wheel. Then the axle was connected to the bike chain which turned with the pedal. When the pedal was turned the wheel also turned, as needed! The same procedure was used for the front wheel before it was connected to the chain. To give the wheel a better grip to the surface of the road, a rubber material was placed along its edges. The wheels were then painted and were ready to roll!
Road of catenary curves 
The road required for a smooth ride can be determined as the solution to a differential equation. The solution is found as a catenary curve. A catenary is the curve made by the gravitational influence on a hanging chain. A classic example of a catenary curve is the Golden Gate Bridge. Now in order to construct a road of inverted catenary curves, we had to choose a material. We decided to construct our road out of wooden "logs". To build a log, catenary curves were cut from 10-foot 2x6 boards. A log is constructed by assembling 20 of these pieces together. The resulting logs have a length of 3 feet and a height of approximately 3.5 inches. Since the road construction was the most labor intensive part of the challenge and the most expensive component of the project, we initially built 11 logs. Later, we built an additional 5, leaving us with a 20-foot road. After our logs were constructed, we covered them with vinyl to add some aesthetic appeal.
More of the math
A catenary curve is a hyperbolic cosine function. The hyperbolic functions are derived by removing the imaginary number i in the complex exponential formulations of trigometric functions. The square wheel on our bicycle has sides of length 17 inches. The smooth ride then enforces that the curved length of a log is also 17 inches. Using techniques of differential equations, multi-variable calculus, and trigonometry, the mathematics provides that the height y of our road (as a function of the horizontal distance x directed toward the forward motion of the bicycle) is:
Engineering hurdles 
The square-wheel bicycle sits lower to the "ground" because the required ground is a curved road, which is elevated from the usual flat road in nearly all places. On the first trial ride, the pedals on the original bike hit our road and we went nowhere. This aspect was overlooked! We then replaced the crank (pedal) set with a child’s set that offered shorter pedals. After some practice, it was found that one of the pedals must be in an upright position (45 degree angle) when starting in order to gain early momentum. Remember, our road ends after 20 feet! As for brakes, the coaster brakes do not allow us to pedal in reverse. Thus, the bike must be picked up each time the pedal needs to be repositioned. As for some maintanence, the chain on our original "discount" bike was rusty and needed to be replaced. Lastly, an additional challenge, although only with respect to the set-up, is that the road logs are heavy and require people and time to transport them.
Proof by example
At the Fall 2007 Pi Mu Epsilon Regional Undergraduate Mathematics Conference at St. Norbert College, which has been held annually since 1985, the Mathematical Modeling class in self-designed T-shirts debuted the square wheeled bicycle to an anxious crowd. The outcome? Applause and cheers. It works! Check it out.
What's next? 
Our project has generated a lot of enthusiasm in a variety of venues. We continue to learn new bicycle tricks and are even considering new wheel shapes and roads. The mathematics behind constructing a wheel that travels smoothly along a saw-tooth road has already been explored. We have given lectures at local and national conferences on the mathematics behind each of these problems.
Acknowledgements
The project required patience, creativity, and collaboration. We offer special thanks to the…
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Computer Science Program at St. Norbert College for the work space and tool use.
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SNC Math Department for funding for the project.
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Andrew Wiesner for supplying us with advice on the mechanics behind bicycles.
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Brian Leiterman for giving us his time in helping to construct the road.
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Pi Mu Epsilon Conference for offering a great forum for our bike's debut.
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Whole St. Norbert College community for their support , enthusiasm, and encouragement.
Course description
MATH 489 explores the fundamental concepts and techniques of applied mathematics through the construction and analysis of mathematical models for population dynamics and mechanical vibrations. The mathematical skills developed will involve difference equations, ordinary differential equations, non-linearities, equilibrium solutions and stability, linearization, and phase plane analysis. An essential reason for studying mathematics lies in its applications. By investigating two natural models in physics and biology, students have an opportunity to develop and process mathematics by making observations, carefully formulating a model, obtaining a solution, and then interpreting the validity of that model in an attempt to make reasonable predications about a physical system. The course seeks to enable students to make connections and form relationships between mathematics, models, and science.
A Note from the Instructor
As a relatively new faculty member, I was committed to finding strategies to better manage my teaching, research, advising, and service responsibilities. I am committed to mathematics research which includes both mentoring students on projects as well as maintaining an active research agenda of my own. In Spring 2007, my discipline and I coordinated a rotation of courses that would make room for a new course in mathematical modeling. This is my main research interest. I proposed offering a course under our special topics designation before formally adding it to our curriculum so that I could iron-out the details. In the end, MATH 489 was added to the Fall 2007 schedule.
Besides piquing student interest in mathematical modeling, I used this course as a platform to offer a non-routine laboratory experience. I was inspired by Ken Moffet's YouTube Video where he discusses and demonstrates a "revamped" version of Stan Wagon's original square wheeled tricycle. As the project began, the students respected my request to not use the internet as a resource for the mathematics and engineering. Thus they did not know about Wagon's tricycle until after the construction of their bicycle was well underway.
Although this was a class project, I was quite conscious of the fact that it was part of a course that had plans to be formally included in the mathematics curriculum. For that reason, the classroom was reserved for lessons on the conventional techniques of mathematical modeling. Hence, nearly all the work on the bicycle happened outside of the classroom and always included my attendance.
During the summer before enrollment opened, I contemplated many strategies for motivating students to take on such a task. Certainly one way to get students to participate is to involve the task in their grade. After thought and consultation, I found no fair way to evaluate participation in a project of this kind. So I decided it would not be included in their course grade ... at all. Instead I chose to put faith in an assumption: my students genuinely want to learn and mature intellectually. It may be a leap for some but by making the choice to instruct without any hint of doubt in this, I could solidly encourage the students with the incredible opportunity to do the "impossible." Proof that one can do hard things is something to attain in college and I energized students by asking them to prove of themselves exactly this. And, they did!
Comments from Students
"I honestly didn't think we were going to pull it off. This changed my view on how mathematicians can collaborate to make something or solve something. Having an appreciation and understanding of mathematics makes you look at the world differently then someone who doesn't."
"This project was something FUN! Very rarely have I done a project in a class that was this enjoyable yet also this challenging. I realized that if a question arises, look for the answer! Even if you don't have it, someone might or it might be a discovery."
"I felt like at first I didn't get into the project because I looked at it as another thing I had to do, but when I took the chance and got involved, it became a little like 'my baby' and we were all co-parents. It felt good to claim something and own it."
"The bike project has excited my interest in mathematics to a great extent. I have developed a new way of thinking, this mathematical way of thinking, and I absolutely enjoy it."
Class Participants
Dr. Terry Jo Leiterman, Assistant Professor of Mathematics
Luis Altamirano, Physics, '08
Alicia Brinkman, Mathematics Education, '10
Matt Captaine, Physics and Mathematics, '08
Brandon Clemens, Mathematics and Physics, '09
John Cremer, Mathematics and Business Administration, '09
Dr. John Frohliger, Associate Professor of Mathematics
Kate Kaminski, Mathematics and Business Administration, '09
Kathleen Miller, Mathematics, '10
Ryan Pavlik, Mathematics and Computer Science, '09
Justin Pierce, Mathematics and Computer Science, '10
Stephanie Schauer, Mathematics Education, '10
Heather Schulze, Mathematics and Spanish, '09
Jenni Wirth, Mathematics, '07
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MATH 489 - Fall 2007
Dr. Terry Jo Leiterman
Special Topics: Mathematical Modeling
A Note from the Instructor
Comments from Students
Photo Gallery
Live Video Demonstration
Media Attention
AMS Math Digest
Chronicle of Higher Education
@St. Norbert, December 2007
Green Bay Press Gazette
"Conversations from St. Norbert College" television show, January 2009
Exhibitions
November 2-3, 2007
Pi Mu Epsilon Regional Undergraduate Math Conference, SNC
April 8, 2008
Celebrating Student and Faculty/Staff Collaborations, SNC
April 12, 2008 Big Event for Little Kids 9-3 p.m. Shopko Expo Center, Green Bay, WI
April 24-25, 2008
Wisconsin Section MAA Meeting, Madison Area Technical College, Madison, WI
August 7, 2008 Math Fest, Madison, WI
February 24, 2009 MATC Math Club Lecture Series, Madison, WI
April 4, 2009 Big Event for Little Kids 9-3 p.m. Shopko Expo Center, Green Bay, WI
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