Square-Wheeled Tricycle

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Square-Wheeled Tricycle
Square Wheeled Trike.jpg
On display at National Museum of Mathematics
Type Hands-on, playground
Topics Plane curves

The Square-Wheeled Tricycle is an iconic exhibit at MoMath, where the visitor can ride a tricycle with square wheels on circles over a special track so that the ride is smooth.


Two tricycles with square wheels ride on a circular track made of inverted catenaries. The three wheels of the tricycle have different sizes, depending on the distance between the wheel and the center of the circle track (that is, the radius of the circle created by that wheel’s path around the center). The front wheel of the tricycle has pedals that the visitor can use to power the vehicle. The two rear wheels are connected through a gearbox, which makes the two axles of the wheels parallel (but not at the same height, to compensate for the different sizes of the wheels), and to lock the wheels together, keeping them in sync.

Activities and user interaction

Visitors delight in taking a smooth ride on a vehicle with square wheels. The two trikes are very different in size, allowing for large adults and small children to participate. The two trikes, which can be ridden simultaneously, move in opposite directions, creating a carnival like movement. Friends and family members spontaneously high-five each other as the trikes pass through the same section of the track.

Mathematical background

A square wheel with the axis of rotation at its center can rotate over a special track, making the rotation axis stay at constant height. In order to achieve this, the track must have the shape of a sequence of inverted catenaries with appropriate size. The equation for a catenary is y = 1/2 (e^x – e^(-x)), alternatively known as cosh x. This is also the shape one gets when one lets a chain or a string droop between two endpoints.

The difference in wheel sizes provides an opportunity to discuss the relationship between the radius of the wheel’s path and its circumference.

History and museology

The Square-Wheeled Tricycle is based on the work of Stan Wagon, a mathematics Professor at Macalester College. Stan Wagon’s vehicle rides on a straight track, requiring the trike to be stopped and reversed at the end of the track. MoMath’s circular track provides continuous movement with no resetting needed.