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. If you choose to take your own tour of London and want to make it a bit more interactive, look at the ‘Demonstration’ sections listed in the full tour to see what materials you need to bring with you.
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 Millenium Bridge, London
6. The dome of St Paul's cathedral, London
7. Topology on the Tube
You can download a printable copy of the tour.
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.
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 contains 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 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 ofgraph 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.
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.
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.
Euler's solution demo
You might like to try out Euler's solution on a network created by the group. Ask 8 (or so) people to create a network by taking the ends of two or more of a dozen (or so) pieces of rope that are spread out between them. Then ask everyone to check how many pieces of rope they are holding – as long as all of them have an even number of ends, or exactly two of them have an odd number of ends, then Euler's solution says that you should be able to move through the network travelling over each link (piece of rope) exactly once. Try to navigate the resulting network (it works well to pass a small hoop over each piece of rope) placing each piece of rope down as you pass over it, therefore ensuring you only use each link once.
· 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
· Euler's solution demo
· 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.
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.
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 paralleogram. 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.
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 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 calledparallelepipeds.)
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.
Lavender Mist by Jackson Pollock
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!
5. The Millennium Bridge, London
There was great excitement when the Millennium Bridge opened in June 2000. Thousands of people flocked to walk across the first dedicated pedestrian bridge built over the Thames since 1894. But as they started to make their way across it the bridge began to sway, affecting people's walking patterns and earning the bridge's notorious nickname, the Wobbly Bridge. But what caused this wobbling?
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 foot steps 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, freqencies, 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 synchronsing 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.
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.
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 freqency, 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.
· 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.
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 architecure 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 caternary 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 succesively smaller terms.
[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.
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.
Props: Several laminated copies of Wren's sketch of the triple dome, available below (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.
· 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.
Oh what a tangled web...
As you are in London, you are almost certainly aquainted 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) everyday 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 corrrect 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 groundbreaking 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.
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!
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!
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.