Maths Goes Underground - Institute of Mathematics and its

Maths Goes Underground
W
ith the 50th anniversary of
the IMA less than a year away,
it is good to celebrate another
anniversary this year, namely the 150th
anniversary of the London Underground.
I suspect that just about every member of
the IMA has at some point travelled on
an Underground train. The first such train
(which was gas lit and powered by steam)
travelled on the Metropolitan line on 10 January 1863. Since then
the London Underground has grown considerably in size, and the
network consists 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 London Underground and
Mathematics. In particular, an important landmark in the history
of the Underground was the creation of the Tube map by Harry
Beck in 1931. Beck recognised that because the railway ran mostly
underground, the physical locations of the stations were irrelevant
to the traveller wanting to know how to get from one station to
another. He simplified the network based on the interactions of
the lines themselves rather than their actual location relative to
one another. The genius behind the creation of the Tube map was
a compression of the essential information into a diagram, which
was clear, informative and had great artistic appeal. For many of
us this is our first (and often only) introduction to the great mathematical subject of Topology. The Tube map is also perhaps the
most famous example of a network, in which the nodes of the
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OCTOBER 2013 198
network 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. It has been emulated widely and serves as a constant reminder of the importance of Topology in
real life!
© Transport for London
The link between the Underground and
labyrinths has led to an inspiring artistic project called Labyrinth
which has been a major feature of the 150th anniversary celebrations of London Underground. Turner Prize-winning artist Mark
Wallinger was commissioned by Art on the Underground to respond to the rich environment and history of the Tube, leading to
a long, considered artistic process which resulted in the creation
of 270 unique labyrinth artworks to be permanently installed 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
A labyrinth is different from a maze, in that there is 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 can be found 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 well
as its close links to mazes and networks, the labyrinth is also a
fitting analogy for the millions of journeys that are made across
the Tube network every day. As Art on the Underground’s website
(https://art.tfl.gov.uk/labyrinth/about/) explains:
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 traveller’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. The picture below shows you how to draw what is often
called the ‘classical labyrinth’. Try it yourself, either on a piece
of paper or (better) draw a very large one in the sand with a stick.
The challenge that Mark Wallinger was faced with, was finding 270 unique designs all of which had a striking impact. This
Mark Wallinger, Labyrinth, 2013 © The Artist, Courtesy Anthony Reynolds Gallery,
London. Commissioned by Art on the Underground. Photograph © Thierry Bal.
required a mathematical algorithm. The result has been a great
success fusing Maths, Art and Design.
The labyrinths are currently being installed across the Underground network. Here is one in situ. Can you work out where it is?
I warmly encourage all Mathematics Today readers to visit the
Underground, seek out the labyrinths, and to enjoy this wonderful
fusion of Maths and Art.
Happy 150th London Underground.
Chris Budd CMath FIMA
Note
1 The Tube Challenge is a race to pass through all 270 stations on the
network in the shortest time possible. Rules state that participants
don’t 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.
Some questions about the Underground
Q1. What is the only Underground station to contain none of the
letters of the word ‘mackerel’ in its name?
Q2. Which is the only Underground station to have all of the vowels (A,E,I,O and U) in its name?
Q3. Which Underground station has the longest name?
Q4. If you pick two stations at random and travel from one to
another taking the route with the minimum number of changes,
what is (i) the average and (ii) the maximum possible number of
changes that you have to make?
Q5. How many Underground stations begin and end with the
same letter?
Q6. What do Goodge Street, Oxford Circus and Fairlop stations
have in common?
Visit http://art.tfl.gov.uk/labyrinth/ for further information
on the project.
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