A Rolling Square Bridge: Reimagining the Wheel

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Alfred Jacquemot, Thomas Randall-Page, Antonin Slavik, and Stan Wagon

Alfred Jacquemot, Price & Myers, London, UK; <alfred.jacquemot@gmail.com>
Thomas Randall-Page, Architectural Association, London, UK; <tomrandallpage@hotmail.com>
Antonín Slavík, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;  <slavik@karlin.mff.cuni.cz>
Stan Wagon, Macalester College, St. Paul, MN, USA; <wagon@macalester.edu>

Mathematics has always played a strong role in bridge design and construction, but there is always the possibility of a radical innovation. A new bridge in London is an example as it uses classic geometry in a very surprising, even shocking way.

The idea of a rolling square portal came from a basic knowledge that gears can be noncircular provided the offset from their pivot was mirrored from one gear to its partner. In exploring this idea the designer came across constructions, such as Wagon’s square-wheel device, that forced the center of mass to move in a straight line. We are used to shapes such as squares and rectangles in our built environment and we are used to wheels being round. A rolling square is therefore unexpected and potentially unsettling (Fig. ). Another factor was the hope that the rolling of the bridge would be performative and dramatic: not something one endured, but rather something that one eagerly anticipated.

1.  Introduction

Cody Dock, near where the River Lea empties into the Thames in East London, is being brought back into use after years of dereliction (see Fig. XX1 -- a map). Thanks to designer Thomas-Randall Page and structural engineer Alfred Jaquemot, a remarkable new bridge over the waterway illustrates a mathematical principle that had never before been used in large-scale construction.

In 1849, at the age of 18, James Clerk Maxwell [, p. 537] discovered that a straight line will roll smoothly on an inverted catenary. The idea was extended in 1960 by G. Robison [], who showed that a square will roll smoothly on a series of truncated and linked catenaries. Stan Wagon, in 1997, constructed a full-sized tricycle to illustrate the idea of a smoothly rolling square (Fig. ). That whimsical construction received a lot of publicity. Randall-Page designed the new Cody Dock bridge as a platform attached to two large squares; the structure can be smoothly rolled to an upside-down position that would allow boat to pass beneath it (Fig. ). He and Jacquemot then oversaw the construction so that the inversion of the 26,400-pound structure can be carried out entirely by hand (Fig. , ). There were many details to be worked out to ensure the new idea would work; for example, a large amount of concrete and steel lies hidden in the two upper edges to counterbalance the weight of the bridge deck. Not only does it work, but the bridge won the 2022 Bridges Design Award [].

Insert map here.

The bridge design relies on two 19th century ideas. There was the aforementioned geometrical result of Maxwell. But another factor was the use of a pre-industrial ethic. Before steam engines and electrical power, all moving things required muscle, and thus balanced systems were used, often with counterweights to minimize the energy needed. This history motivated Randall-Page to come up with a balanced bridge that can be moved by hand. Another motivation for the square was that space was limited and so a design was needed that did not occupy land on either bank. This led to a design that utilizes two catenary tracks along the channel, with no supporting structures on the banks.

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There are some surprising subtleties in the translation of a relatively simple geometrical idea to a large-scale construction. To ensure that the squares could not slide down the track, a system of gears and pins is used. These force the corners of the squares to be rounded, and that means the pure linked catenaries will not work as the road. One must work out the proper road for a circular arc, and that involves elliptic integrals. The details are discussed in §6XX. 

There are several videos that show the bridge in action; see XX, XX, Parker.

2. An Elegant Equation: x'(θ)=r(θ)

Continue as before.
The section on bridge construction should remain, but be shorter.