The Surprising Utility of Square Wheels

The London Eye, a vast observation wheel with a diameter of 120 metres, on the south bank of the River Thames, opened in 2000. Resembling an enormous bicycle wheel, it has 32 passenger capsules, each weighing ten tons, attached to its circumference. Each capsule holds 25 people, so up to 800 may be on the Eye at any one time. The wheel, supported on one side only by an A-frame, spins on a horizontal axis, once every half hour. This is slow enough for passengers to step on and off while the wheel continues to move. The Eye has become one of the most popular tourist destinations in London  [TM264 or search for “thatsmaths” at irishtimes.com].

Squaring the Circle

Among the great problems of classical mathematics was the squaring of a circle. The challenge was to construct a square with area equal to that of a given circle. Within the confines of Euclidean geometry, only a ruler and compass may be used, and the solution must be exact. After thousands of years, Ferdinand von Lindemann proved in 1882 that the number we call pi is transcendental. This implies that the ancient goal of squaring the circle is impossible.

Of course, engineers, whose tools are not limited to ruler and compass, have no problem constructing a square of any size. But who would wish to construct a square wheel? It would certainly not serve for the London Eye. Surprisingly, square wheels are both interesting and useful. If you visit the Museum of Mathematics in New York, you may take a ride on a square-wheeled tricycle around a circular track equipped with humps specially designed to smooth out the ride.

We know how bumpy a bike ride on a rough road can be. It would seem that square wheels would make things worse but, as shown by mathematician Stan Wagon, if the road is formed from a series of arches called inverted catenaries, a smooth ride is possible: the sides of the squares maintain contact with the arches in such a way that the centres of the wheels move horizontally.

Cody Dock Rolling Bridge

The square wheeled tricycle inspired the architect of a bridge located in East London: near where the River Lea flows into the Thames, a footbridge in the form of a large cubic frame straddles the entrance to Cody Dock. Normally, the footway of the bridge blocks access to the inner basin, but it can be rolled through 180 degrees to allow barges to pass underneath. The square frame of the bridge rolls on an ingeniously designed track in such a way that the centre of mass remains at a fixed height. Thus, the twelve-ton bridge is easily operated by a hand-winch. As the corners of the frame are rounded off, some advanced mathematics involving elliptic integrals was involved in the design of the track.

Videos of the square-wheeled tricycle and of the Cody Dock rolling bridge can be found on You Tube:

• Square-wheeled Trike.
• Cody Dock Bridge

The invention of the wheel was a landmark in human development. Perhaps a frivolous observation may be forgiven in this festive season: the brilliance of the inventor of the first wheel is not in doubt, but is overshadowed by the genius of the guy who invented the other three. Happy Christmas to all.