SIAM News Blog

Maths Goes Underground

By Chris Budd

As the UK’s Institute of Mathematics and its Applications looked forward to its 50th anniversary (in 2014), Chris Budd reminded readers of the IMA’s Mathematics Today of a maths-related anniversary that fell in 2013: The London Underground’s 150th. The following article is an adapted version of Budd’s IMA article.*

The first London Underground train (which was gas lit and powered by steam) traveled on the Metropolitan line on January 10, 1863. Since then, the Underground has grown considerably: It consists today of 270 stations joined by 249 miles of track. It has inspired other metro systems all over the world.

There are many close links between the London Underground and mathematics. In particular, an important landmark was the creation of the Tube map by Harry Beck in 1931. Because the railway ran mostly underground, Beck recognised that the physical locations of the stations were irrelevant to a traveler wanting to know how to get from one station to another. Accordingly, Beck simplified the network based on the interactions of the lines themselves, rather than their actual locations relative to one another. Beck’s genius in creating the Tube map was to compress the essential information into a diagram, which was not only clear and informative but also had great artistic appeal (see below.)

For many travelers on the Underground, this is their first (and often only) introduction to topology. The Tube map is also a famous example of a network, in which the nodes are the stations and the edges the train connections between them. The Tube map helped to make complete sense of the complex system of lines in the Underground system. The map has been emulated widely and serves as a constant reminder of the importance of topology in real life!

The mathematics of labyrinths led to an inspiring artistic project, called Labyrinth, that has been a major feature of the 150th-anniversary celebrations of the London Underground. For the project, Art on the Underground commissioned Turner Prize-winning artist Mark Wallinger to respond to the rich environment and history of the Tube. In a long, considered artistic process, Wallinger created 270 unique labyrinth artworks for permanent installation, one in every station on the network. Each artwork has its own reference number, acknowledging the order in which its station was visited during the 2009 Guinness World Record Tube Challenge.1

Figure 1. Step by step, the “classical labyrinth” takes shape.
A labyrinth is different from a maze, in that it has only one route to the centre and out again, although that route may be very long. Labyrinths have a history that can be traced back 4000 years, and they appear in many cultures, the most famous example being in the mythological story of the Minotaur on the island of Crete. Labyrinths are thought to be associated with ceremonies involving dancing and movement. They were also used in defensive structures, such as Maiden Castle, where the attackers were forced to trace a very long route to the entrance, during which time they were under constant attack!

Why the labyrinth as the theme of the anniversary art? Along with its close links to mazes and networks, the labyrinth is a fitting analogy for the millions of journeys made across the Tube network every day. As explained on Art on the Underground’s website:

“Rendered in bold black, white and red graphics, the artworks are produced in vitreous enamel, a material used for signs throughout London Underground, including the Tube’s roundel logo, whose circular nature the labyrinth design also echoes. Positioned at the entrance of each labyrinth is a red X. This simple mark, drawing on the language of maps, is a cue to enter the pathway. The tactile quality of the artwork’s surface invites the viewer to trace the route with a finger, and to understand the labyrinth as a single meandering path into the centre and back out again—a route reminiscent of the Tube traveler’s journey.”

The mathematical interest follows from the fact that a labyrinth can be created from a basic seed, followed by the application of a set of systematic rules. The design and classification of all possible labyrinths leads to many interesting mathematical questions. The long paths possible within a labyrinth are excellent examples of space-filling curves, which are themselves closely linked to fractals. Figure 1 illustrates the steps for drawing what is often called the “classical labyrinth.” 

Labyrinth, 2013, one of 270 unique artworks created by the artist Mark Wallinger for installation in the stations of the London Underground to mark its 150th anniversary. © the artist, courtesy Anthony Reynolds Gallery, London. Commissioned by Art on the Underground. Photograph © Thierry Bal.
The challenge facing Mark Wallinger was to find 270 unique designs, each of which would have a striking impact. This required a mathematical algorithm. 

The labyrinths are currently being installed across the Underground network, with about 253 in place so far. The photo to the right shows one in situ. 

I encourage readers who pass through London to visit the Underground, seek out the labyrinths, and enjoy this wonderful fusion of maths, art, and design.

Visit for further information on the Labyrinth project.

1 The Tube Challenge is a race to pass through all 270 stations on the network in the shortest time possible. The rules state that participants do not have to travel along all Tube lines, but must pass through all stations on the system. They may connect between stations on foot or by using other forms of public transport.

*The original appeared in Mathematics Today, 49:5 (2013), 198–199.

Chris Budd, CMath FIMA, is a professor of applied mathematics at the University of Bath and a professor of mathematics at the Royal Institution of Great Britain.

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