Maths in the City: London Description Tour steps 1. Graphs ... tour... · another time and place,...
Transcript of Maths in the City: London Description Tour steps 1. Graphs ... tour... · another time and place,...
Maths in the City: London
This is the route and notes to accompany our Maths in the City tour of London. These notes
can also be found on http://www.mathsinthecity.com/tours/maths-city-london.
Description
This is the Maths in the City walking tour of London. We'll be looking at networks,
geometry, resonance, curves and topology through the medium of chalk, sweeties, slinkies
and rope!
The tour is suitable for anyone of any age and includes a lot of demonstrations that illustrate
the maths behind what you see.
Tour steps
1. Graphs and networks at the Tate Modern, London
2. Creating the Gherkin's curves, London
3. Art at the Tate Modern
4. Catenary chains on the Thames
5. The Millennium Bridge, London
6. The dome of St Paul's Cathedral, London
7. Topology on the Tube, London
1. Graphs and networks at the Tate Modern, London
As we start our walking tour of London, we cast our minds back to a walking tour over
another river that gave birth to a new area of mathematics.
Sumner Street, Bankside, London SE1 9TG
Viewpoint:
In front of the Tate Modern by the river.
The Tate Modern Gallery itself is open Sunday – Thursday, 10.00–
18.00 and Friday and Saturday, 10.00–22.00. But the bankside area is
open at all times.
The bridges of Königsberg
We are not the first people keen to go on mathematical tours of a
city. As we start off on our mathematical walking tour of London,
standing here on the banks of the Thames, let’s cast our minds back to
another time and place, to Kaliningrad, a city in the Russian
enclave between Poland and Lithuania, and to 1735 when that city was part of Prussia and
was known as Königsberg.
Königsberg, like London, was divided by a river. The city was arranged on the north and
south banks of the River Pregal, which widens to contain two islands. The banks were linked
to the islands by seven beautiful bridges. The story goes that the intellectuals of the time
liked to play a parlour game when they met in the coffee houses or salons: could you come up
with a walking tour of the city of Königsberg that crossed each of the seven bridges once and
only once. [Can you suggest a path they could have taken?].
Euler’s solution – a new map of the city
The people of Königsberg could not seem to
come up with the winning route. But the
mathematician, Leonhard Euler, was able to
solve the puzzle.
In Euler’s new map he ignored the geography,
the types or lengths of the bridges, and instead
presented each piece of land as a circle linked
by a line if they were joined by a bridge. With
this new perspective Euler was able to quickly
solve the problem: there is no route that
crosses each bridge of Königsberg once and
only once. [Does anyone know why?]
Euler’s new type of map is called a graph, which is simply a set of nodes (the pieces of land)
that are connected by links (the bridges). Euler realised that a route that crosses each bridge
exactly once is only possible if either exactly zero or two of the nodes have an odd number of
links. And all the nodes on the graph of the bridges of Königsberg have an odd number of
links – therefore there is no walking route that crosses each bridge once and only once. [Why
is such a route possible only if either exactly 0 or 2 nodes have an odd number of links?]
This is because if you don’t want to get stuck on a piece of land on the way, each one has to
have an in-bridge and out-bridge, which would give it an even number of links. The only
possible exception is that you end in a different place than you started, in which case the
beginning and ending points have an odd number of links.
Graph theory and network science
It is very rare to pinpoint with precision the moment a new field of mathematics was
born. But when Euler discovered this solution to the Bridges of Königsberg problem in 1735
he founded the field of graph theory. [Can anyone think of something else you could
represent by a graph, a set of things that are linked together? What are the nodes? What are
the links?]
Graphs are used to map any network, including those that criss cross our cities: transport
networks of roads, train lines and bus routes; social networks of people linked by family and
friendships; and power grids with buildings linked by power lines to power stations, such as
the Bankside Power Station behind us, that now houses the Tate Modern art gallery.
Mathematically describing a network allows us to better understand how things can move
around the network, for example how traffic flows around the city. Traffic congestion isn't
just a modern problem. Albemarle Street became the first one-way street in London when
horses and carriages caused traffic jams as crowds flocked to the science lectures at the Royal
Institution in the nineteenth century.
Network science is hot topic of research and brings together many disciplines. Mathematical
discoveries, made just in the last decade, are revolutionising the way we study the spread of
diseases and altering the way we fight diseases such as AIDS.
Demonstration
Props: chalk, sweets, a dozen pieces of rope (each around a metre long), a small hoop to pass
over the pieces of rope (a ring-shaped frisbee works well)
We've included questions (in italics in the description above) that you can ask the group to
get them involved and a crib sheet below to suggest how to interweave the demos into your
explanation.
Bridges demo
Before you start the tour, chalk the layout of Königsberg and its bridges on the ground so that
people can try out their walking route. If you put something on each bridge (a sweet is a
good incentive!) they can keep track of their route by only crossing bridges that contain a
sweet, collecting the sweets as they go.
Then you can draw out Euler’s graph alongside your city layout so that the group can see
how the pieces of land correspond to the nodes and the bridges to the links. This diagram of
the graph will also make your explanation of Euler's solution clearer.
Crib sheet
Chalk out layout of bridges of konigsberg before hand
Explain Bridges of Königsberg problem
Bridges demo
Explain Euler's solution, chalking out graph next to the bridges layout
What is a graph, node and link
Other examples of networks and impact of new understanding
2. Creating the Gherkin's curves, London
You can’t take a tour of London and not notice one of the most iconic elements of the skyline
– 30 St Mary Axe – otherwise known as the Gherkin. The design and construction of this
striking building would not have been possible without mathematics. In fact its tapered
curved shape and spectacular construction all rely on the strength of the humble triangle.
Viewpoint:
River bank, just south of the Millennium Bridge
Iconic curves
Look along the opposite river bank. What shapes do
you see? Although most buildings are based on
rectangles and squares, two curved icons dominate the
skyline: the spherical dome of St Paul's Cathedral and
tapering curved shape of 30 St Mary Axe, aka the
Gherkin. [Note this view is now obscured. Show
pictures of Gherkin instead.]
Creating enormous, curved buildings is a far more
difficult task than one based on vertical straight
lines. To ensure a spherical dome will support its own
weight it needs to be built with a thickness of at least
4% of its radius. The dome of St Peter's, Rome, has a thickness of about 15% of its radius.
Wren used a different approach in his construction of the dome of St Paul's as we'll see a little
later in this tour. But what if you want to build a much larger curved form out of the thinnest
possible shell, made of just steel and glass?
The strength of triangles
The answer lies in that most uncurvy of shapes: the triangle. [How many pieces of curved
glass are used in the Gherkin?] Although the Gherkin appears to have a sleek curved form it
is actually made up of hundreds of flat panels, with just a single curved piece of glass right at
the very top of the building. Not only do these panels give the iconic building its curved
appearance, the strength of their triangular shape also allows the building to have such a light
and airy structure.
[Why are triangles so strong?] Triangles are inherently strong because they form a fixed
rigid shape. This can be demonstrated by building a triangle out of garden canes, securing
the corners with rubber bands. The shape is fixed by the length of the sides and the triangle
withstands quite substantial forces applied to it.
[Why is a square unstable?] However if you built a square in the same way, a gentle push at
one corner could easily change the shape into a parallelogram. There are infinitely many
four-sided shapes with equal sides, a square is just one of them, and so the shape is easily
transformed from one to the other with minimal force.
A square lacks the rigid strength of a triangle. But by adding diagonal bracing, a common
feature in bridges and buildings, the structure can again rely on the strength of a triangle to
hold its shape.
Creating the new world
Triangles have always been a fundamental tool in architecture and construction, but
mathematics has allowed them to be used to construct some of the complex and daring shapes
we see in architecture today. [Can you spot any other triangles along the river or in the
city?] Triangulation of surfaces, such as the Gherkin or the roofs of the British Museum
courtyard and the new concourse at Kings Cross Station, has allowed spectacular curved
shapes to be built with a minimal amount of material.
Triangulation of surfaces is also responsible for the virtual worlds many of us enjoy. Digital
images and computer generated animation all rely on clever mathematics to construct and
manipulate triangulated models of our favourite characters and their environments, and then
to mathematically paint them to bring them to life.
Demonstration
Props: 20 garden canes and lots of rubber bands.
We've included questions (in italics in the description
above) that you can ask the group to get them involved.
Divide the group into two groups and ask one to construct
a tetrahedron (a pyramid made out of four equilateral
triangles) and the other a cube.
When finished, ask the first group to let go of their tetrahedron. The structure will stand
unsupported due to its rigid shape and will even bear a substantial amount of force. This is
because there is only one possible shape that can be constructed with four triangular faces
with equal length sides.
The carbon atoms in diamond are arranged in a continuous lattice of tetrahedra. This is why
diamonds are so hard.
When the second group let go of their cube, the structure will collapse. The shape isn't rigid
as the corners are flexible. The shape can easily be skewed into any one of the infinitely
many shapes that can be built with six quadrilateral faces with equal length sides. (These
squashed cubes are called parallelepipeds.)
3. Art at the Tate Modern
A beautiful example of maths hangs in the Tate Modern. The famous drip paintings of
Jackson Pollock have lines of paint that seem to fill the canvas and, no matter how close you
look, the painting appears the same: they have a fractal structure. Mathematical analysis has
even been used to distinguish Pollock's genuine paintings from forgeries. Chaotic pendulums
can mimic Pollock's physical method but we are yet to automate the innate aesthetic
judgment of the artist's eye.
4. Catenary chains on the Thames
The shape formed by a chain hanging under its own weight suspended from either end is
called a catenary curve. This shape plays a vital role in architecture as it is the perfect shape
for an arch. There are some lovely examples of this on the walls of the Thames!
Millennium Bridge, London, UK
The wobbly bridge
Suspension bridges always wobble a small amount; this is simply down to their design.
Normally, this causes no problem as individual wobbles are not allowed to build upon one
another and, on the whole, people tend not to be synchronised.
However, as the crowd began their maiden voyage across the bridge an unexpected effect
began to emerge. The people were synchronising their footsteps with the movement of the
bridge, causing it to wobble more and more. The reason behind this synchronisation is that, as
the bridge began to wobble a small amount, the crowds unconsciously changed their pattern
of movement to counteract it. However, instead of reducing the motion, they were causing a
feedback loop, in which the more they tried to stop the motion, the more the bridge would
wobble (you can see this in the youtube footage).
Resonance and natural frequencies
The reason this feedback loop was so problematic was that the natural resonant frequency for
the sideways movement of the bridge was so closely matched to the average person's walking
pace (which is about 1.7 Hz, or two steps per second).
Objects most readily vibrate at their natural frequencies. [Does anyone play a musical
instrument?] Musical instruments are a clear example of the importance of natural
frequencies and resonance. For example, when you pluck a string on a guitar it resonates
most strongly at the fundamental frequency, where the string oscillates between the two fixed
ends.
However, strings also resonate at other, higher, frequencies, called the harmonics, which are
multiples of the fundamental frequency. [Does anyone know what the second harmonic is
called?] The second harmonic, known as the octave in music, has twice the frequency of the
fundamental. [Does anyone know what the third harmonic is called?] The third harmonic
has three times the frequency of the fundamental and, musically, is a perfect fifth above the
octave. (The opening two notes of the Star Wars theme and Twinkle, Twinkle, Little Star are
examples of perfect fifths).
It only takes a little push...
If you've ever pushed someone on a swing you'll know that it doesn't take much effort to
make them swing very high, as long as your pushes are in time with their swinging. This is
exactly what happened when the Millennium Bridge first opened.
Like all suspension bridges the Millennium Bridge is flexible and swayed a little in response
to wind and foot traffic. What was surprising, however, was that people reacted to this slight
movement by gradually falling into step with the sideways motion of the bridge. This seems
to have happened because the bridge naturally swayed at a frequency that was close to the
average walking pace for people crossing the bridge, around 1.7Hz.
As people synchronised their steps in response to the bridge's movement, they each provided
an additional push in time with the sway of the bridge. This in turn increased the distance the
bridge swayed from side to side (in exactly the same way as a small push in time with a
swing will make it go higher and higher) which then led to more people synchronising their
steps. This feedback loop continued until the movement became so large that people began
to stop walking and hold onto the hand rails for support.
Very quickly the bridge was closed and stabilisers were installed to stop the motion from
occurring. You can see these beneath bridge as you walk under it on the south bank. [Can
anyone see the stabilisers?]
Dangerous yet beautiful
[Can anyone suggest another situation where resonance can be dangerous?] Unexpected
resonance can be dangerous, whether for bridges exposed to wind and traffic or for buildings
responding to the vibrations of earthquakes. However, resonance is also a thing of beauty. It
is precisely the resonance of musical instruments that creates their beautiful sounds.
And these mathematical harmonies have also been important ideas in science and
architecture. Palladio, the sixteenth century Venetian architect, is said to have created "frozen
music" as his buildings had proportions that mirrored these harmonic ratios. There is a
beautiful example of this frozen music just down the river in Greenwich – the Queen's House
built in the seventeenth century by Inigo Jones was the first example of Palladian architecture
in Britain.
Demonstration
Props: A long slinky toy or rope, 4 heavy metal washers, fishing line or fine string.
We have included some questions in italics in the text above to help engage the group and a
crib sheet below to suggest how to interweave the demos into your explanation.
Natural frequencies demo
Use the rope/slinky to demonstrate standing waves that are created when you vibrate the rope
at the fundamental or harmonic frequencies. Get one person to hold the rope still at one end
and the other person to shake it up and down. (If you are using a slinky get them to hold onto
the first few coils at either end). See if they can generate the fundamental frequency first,
then, as they increase the speed of their vibration, the second and third harmonics should
appear. See http://www.mindbites.com/lesson/4603-physics-in-action-standing-waves-on...
for a good demonstration of this.
Feedback loop demo
Use the washers and line to make four pendulums, two identical long pendulums and two
identical short pendulums, all suspended from another line. Hold the suspension line very
taut and make all the pendulums hang still. Then start one of the longer pendulums swinging
and you’ll see that the other long pendulum starts moving with a similar frequency, while the
two shorter pendulums stay relatively still.
This is because even the small movements from the swinging pendulum, transmitted by the
suspension line, will kick the other long pendulum into motion as these pushes are at the
natural frequency shared by both pendulums. However as the natural frequency of a
pendulum is determined by its length, these pushes are not at the resonant frequency of the
shorter pendulums, hence they remain relatively still. You can see a video of this
demonstration on YouTube.
The movement passes between each of the similar pendulums, illustrating the feedback loop
that occurred when the Millennium Bridge first opened, where the movement of the bridge
affected the movement of the people, which affected the bridge, and so on.
Crib sheet
Intro: problems with the bridge when it first opened (if using headsets you can start
introducing this site as you walk over the last third of the bridge)
Problem was that natural frequency of sideways movement of the bridge was very
close to the average walking pace
Natural frequency demo using the slinky
If you've ever pushed someone on a swing...
Feedback loop demo with pendulums
Slight movements of bridge, caused people to walk in step, caused more movement in
bridge...
Unexpected resonance can be dangerous
But resonance can also be beautiful, for example musical instruments and Palladio's
frozen music
6. The dome of St Paul's Cathedral, London
One of London's most loved landmarks, St Paul's Cathedral, has looked over the city for more
than three centuries. And hidden within its dome is an intriguing example of the interplay
between maths and architecture.
St Paul's Cathedral, St Paul's Church Yard, City of London, EC4M 8AD, UK
Viewpoint:
On the river side of the road, facing St Paul's, just near the tourist information. The
Cathedral itself is open Monday to Saturday 8.30am - 4pm
It is hard to imagine London without St Paul's
Cathedral. But, in fact, this is not the cathedral
Christopher Wren wanted to build. Wren's preferred
design featured sweeping curved walls that would have
made the cathedral England's first baroque building. The
curves of this dynamic design were typical of baroque
architecture and echoed the fascination with movement at
that time in science. (The massive and intricate model
Wren made of this design, known as the Great Model, is
displayed upstairs in the Cathedral.)
Although his baroque design for St Paul's was rejected –
considered to be too foreign and not practical enough – its
crowning dome did survive and has now looked over the
city for more than three centuries. And this dome not only
reminds us of his inspiration from foreign architecture, it
also reveals his love of mathematics.
Today Christopher Wren is best known for his architecture. In fact it was mathematics that
Wren studied at the University of Oxford and Isaac Newton regarded him as one of the
leading mathematicians of the day. When he designed the dome of St Paul's Cathedral, Wren
applied the cutting-edge theory of his colleague Robert Hooke about the mathematical shapes
of ideal masonry domes and arches: one of the earliest recorded instances of mathematical
science being used in architecture.
The shape of an ideal arch
Hooke understood that the tension passing through a hanging chain is equivalent to the
compression in a standing arch. He published his work in 1675 in a Latin anagram (a
common method in the day to protect intellectual property while claiming priority for a
discovery). Unscrambled and translated, Hooke's anagram reads: “As hangs the flexible
line, so but inverted will stand the rigid arch”.
The line of thrust of an arch is the way that the weight
of the arch is distributed through its structure. For an
arch to be stable it needs to contain this line of thrust,
either in the material of the arch itself or in its
abutments. Hooke had discovered that the shape of the
line of thrust for a free standing arch was the inverted
shape of a hanging chain. [Does anyone know what
this shape is?] Therefore the ideal shape for an arch,
the shape requiring the least material, is a catenary
curve.
The shape of an ideal dome
[Can anyone suggest which mathematical curve describes the shape of a catenary?]
Hooke and Wren thought the mathematical description of a catenary curve, the ideal shape of
an arch, was a parabola: y=x2. Therefore they thought the shape of an ideal dome would be
given by rotating positive half of the cubic curve: y=x3. Although neither of these equations
are the correct mathematical description (the shape of a catenary curve is actually given by
the hyperbolic cosine function) they are actually incredibly close. The mathematical equation
for the shape of an ideal dome does begin with a cubic term, x3, followed by a series of
successively smaller terms.
Triple dome
[What shape is the dome of St Paul’s?] The dome of St Paul's cathedral is actually made up
of three domes. The dome that you see from the outside is spherical in shape. Not only is the
sphere an intrinsically beautiful shape, its simplicity and perfection made it an important
symbol for the church and represented the shape of the cosmos.
But the dome you see within the cathedral is not actually the inside of the external dome.
Instead it is a smaller dome in keeping with the internal proportions of the building. And
hidden between this inner and outer dome is a third, more steeply sloped, brick dome that is
one of the first documented instances of mathematics being used to design a building.
Mathematical design
One of Wren’s early sketches for the triple dome design, from around 1690, shows that he
used a mathematical curve to define the shape of the middle dome; the cubic curve y=x3 is
clearly plotted on the axes marked on the design. The curve not only defines the shape of the
middle dome but also the height and width of the surrounding abutments, positioned so as to
contain a continuation of the cubic curve to ground level.
Wren's used a cubic curve to define the shape of the middle dome as he and Hooke believed
this was the ideal shape for a dome. This hidden dome provides an invisible support the
structurally weaker, spherical outer dome and to the heavy lantern that sits above.
Maths in architecture and engineering
This sketch is one of the earliest documented instances of mathematical principles being used
in architectural design. We are accustomed to maths being a vital part of architecture and
engineering today but it is quite amazing to think of such scientific principles being used at a
time when there were no calculators, no computers, no electronic surveying equipment nor
GPS.
But of course maths had already been part of the design and construction process for
millennia. It’s thought that Egyptians used right-angled triangles to guide the placement of
stones in the pyramids, ensuring they constructed these with the correct slope. And stone
masons have used geometry to create the beautiful cathedrals and buildings all around
us. The master builders would mark out the shapes they wanted for elements of the buildings
– the windows, decorative motifs and arches – constructing these shapes using basic
geometric techniques and scratching them onto a whitewashed wall or floor of the
building. Then masons could create templates in wood or stone from these shapes and wash
away the whitewashed markings when they were finished. But sometimes scratches
remained in the stone, suggesting the mathematical nature of their designs.
And, as we saw earlier in the tour, maths is a vital part of architecture today. Large
architectural practices have teams that specialise in mathematically modelling buildings so
that the architects have a chance to experiment with their three dimensional designs before a
single drop of concrete has been poured. Complicated and unusual shaped buildings such as
the Gherkin in London would not have been possible to design, let alone construct, without
the use of maths. So we have maths to thank for the iconic buildings, past and present, on our
city skyline, from the curves of the dome of St Paul’s to the tapering form of the Gherkin.
Demonstration
Props: Several laminated copies of Wren's sketch of the triple dome (say 6 copies for a group
of 15-20), a chain to demonstrate the catenary curve, blocks to build catenary arches. (You
can create a catenary arch by printing out an arch shape, making sure it contains a caternary
curve, and using this shape as a template to create a wooden arch. Then cut this arch into
roughly equal sized blocks, numbering the blocks to make building the arch easier.)
We've included questions (in italics in the description above) that you can ask the group to
get them involved and a crib sheet below suggesting how to interweave the demos into your
explanation.
Arch building demo
Ask two people to build the catenary arch out of the blocks. It will take both of them to hold
the pieces in place until the arch is complete, then the arch should stand unsupported and
even support some extra weight.
Crib sheet
This isn't Wren's original design
Dome reveals his love of maths
Shape of an ideal arch – pass chains around group for them to create the shape for
themselves
Arch building demo
Hooke and Wren thought catenary was a parabola, and so thought ideal dome would
be a based on a cubic
Triple dome of St Paul's
Pass around Wren's sketch showing cubic curve – middle dome based on Hooke's
theory of ideal dome
One of the earliest recorded instances of maths being used in architectural design,
surprising given no calculators, computers, etc.
Maths now a vital part of engineering and architecture today
7. Topology on the Tube, London
Many tourists (and Londoners) love the Tube as it so easy to navigate. But most don't realise
that maths, and some clever design from the 1930s, is showing them the way.
Viewpoint:
In courtyard between tube station and St Paul's cathedral
Oh what a tangled web...
As you are in London, you are almost certainly acquainted with this:
[What is it?] This is a geographically correct map of the London Underground. It's hard to
believe that this mess is the true face of the sensible, orderly Tube network.
One of the reasons millions of people ride the London Underground (the Tube) every day is
that it is an incredibly convenient way to get around. Not only does it cover a good deal of
London, it is also very easy to navigate thanks to an important area of maths.
The map of the London Underground is a brilliant piece of design. It is easy to see how to
get from one station to another as it is clear what lines each station is on and how these lines
are connected. The Tube map is striking, clear and orderly. This achievement is even more
remarkable when you consider the Tube map drawn as it actually lies on the land. The
geographically correct representation above has sprawling tendrils and a twisting knot of
lines in the centre.
Untangling the Tube
The current Tube map is still closely based on the ground-breaking design produced by
Harry Beck in 1933. Beck wasn’t a graphic designer: he was an engineering draftsman
working for the London Underground. He disliked the old, geographically correct map and
so, unasked and in his spare time, he came up with a new design.
Beck realised that in order to navigate the Tube it was most important for travellers to know
what lines stations were on and how these lines were connected. Therefore he overruled the
geography and instead placed stations roughly equally spaced on the lines, which he ran
either horizontally, vertically or at 45 degrees. The result was that he shrank the map down
to fit it into a much smaller space with a design that was much easier to read. By squeezing
and stretching the tube lines Beck fitted this sprawling, twisting map into something you can
fit in your pocket!
Just 500 copies of his map were trialled initially but it was so popular it became the standard
way to depict the London Underground. It is now recognised as a design classic and the
same principles are used for many other transport systems around the world.
Topology
Beck’s map is a great example of the importance of an area of maths
called topology. Distances disappear in topology so that the size or
shape of an object no longer matters – you can stretch or squeeze
something and in topology’s eyes it remains unchanged. What is
important is how things are connected; no cutting or tearing is
allowed. A famous joke is that a coffee cup is topologically the same
as a doughnut as you can smoothly deform one into the other!
Poincaré's Conjecture
A doughnut and a coffee cup might be topologically the same but you will never be able to
turn either of these into a sphere. This is because a doughnut (know mathematically as a
torus) has a hole, where as a sphere doesn't. Understanding which objects have holes is vital
in topology as this affects how things are connected. In 1904 the French mathematician
Henry Poincaré posed one of the most famous conjectures in mathematics – Poincaré stated
that, topologically, the only shapes that have no holes are spheres.
This was might be obvious for objects in three dimensions but the question of whether
Poincaré's conjecture was true for higher dimensions remained unanswered for nearly a
century. Finally, in 2002, the Russian mathematician Grigori Perelman hit the headlines
because not only did he prove that this characterisation was correct, he also refused all
accolades for his incredible mathematical achievement.
From DNA to the Universe
Topology is vital in many areas. It has made huge contributions to biology where it helps to
describe and understand how proteins, DNA and other molecules fold and
twist. Cosmologists need topology to determine the shape of the Universe. And topology is
vital in understanding the structure of graphs in network science, as we saw in our first stop
on this tour.
But of course, sometimes on the ground, distances do matter. If we want to get from here, St
Paul’s, to the Barbican, it's easy to see from the map of the London Underground that the
shortest route is to jump on the red Central line to Bank, change to the black Northern line to
Moorgate and change to the yellow Circle line to the Barbican. What isn't obvious, however,
is that it's actually quicker to walk there and should only take you about 8 minutes! Topology
is vital in understanding the overall structure of the Tube network, but sometimes a little local
knowledge goes a long way!
Demonstration:
Props: laminated pocket copy of tube map, laminated copy of geographically correct map,
four sets of rope handcuffs.
This demonstration is a great way to introduce the concept of topology. Choose two pairs of
people. For each pair (A and B), put person A’s hands in one set of cuffs and then person B’s
hands in another set of cuffs so that their arms are interlinked. The two pairs of people are
linked together in a similar (but importantly not the same!) way to a pair of linked rings. The
aim is to see which pair can be first to separate themselves, without taking their hands out of
their own cuffs or breaking the rope – no cutting or tearing is allowed in topology!
Let the two pairs try to extricate themselves, perhaps giving some clues. If no-one succeeds,
draw the group together for the explanation.
The secret is to recognise that each person has a small gap between their wrists and the
cuffs. If you imagine that person B is shrunk down to the size of a bangle encircling person
A’s wrist (remember shrinking is allowed in topology) then that bangle can just be slipped off
through the gap in the cuff. (You can see an excellent explanation at
http://mathsbusking.com/shows/zeemans_ropes/.)
Topology is the secret to solving this problem, just as it was for Beck’s redesign of the Tube
map – it doesn’t matter what size things are, what is important is how they are connected.