Work in Progress

Category: Research

Bristol Bridges Walk
2026-08-13

Bristol Bridges Walk

Art, Exploration, Networks, Try-it-yourself, Wanderings

I was teaching Engineering Mathematics at the University of Bristol when it happened: “Is that possible in Bristol?” one of my students asked. It was a reasonable question. I had been talking about the Königsberg Bridge Problem, the foundational problem that created Graph Theory and by extension modern Network Science.

The problem asks if one can go on a walk in historical Königsberg that crosses each of the towns bridges exactly once. The answer is no: Before you can complete the walk you will always trap yourself on an island that you can’t leave again without crossing a bridge for the second time.

But was it possible in Bristol? It’s an interesting question. Bristol has a population of six-hundred thousand, very similar to Königsberg, which now goes by the name Kaliningrad. Like Königsberg, Bristol is build around the last crossing on a major river, approximately five miles from the coast. And like Königsberg, Bristol is famous for its bridges: One of it’s main attractions is the Clifton Suspension Bridge, and one of the most notable events in Bristol’s history is the Bristol Bridge Riot. The very name Bristol derives from Brycgstow, Anglo-Saxon for The Place of the Bridge.

Bridgewalk on Youtube

I have made a couple of videos about the Bristol Bridgewalk, in which I explain the theory behind the walk while walking it.

Watch on Youtube

Königsberg Bridge Problem

Leonard Euler’s solution of the original Königsberg problem was so beautiful that it triggered the development of Graph Theory, the foundation of modern Network Science.

If you want to read more about the original problem you can do so in this post.

What’s a Bridge?

Making mathematics relevant to the real world always requires a special glue that we call modelling. Mathematics has clear criteria for the existence of eulerian walks, that is walks that cross ever edge in a graph exactly once. But a graph is a mathematical object and a city with its rivers and bridges is not. Maths in itself does not know what a bridge is, that is for the modelling to decide.

When I tried to decide if the Bridgewalk is possible in Bristol I quickly ran into two problems: What is a bridge? And what is Bristol? You might assume that answering these questions is easy or if not easy it is at most a superficial and profoundly uninteresting. But modelling is an essential part of making maths relevant in the real world. It has its own challenges, but if we take these challenges seriously we often gain a much deeper understanding of the system we are dealing with.

But it’s clear what a bridge is isn’t it, I hear you asking. No, it’s not. Using tools like open street map and satellite imagery it is easy to locate bridges and you find a lot. Some will be major bridges over rivers that are unambiguously bridges. But some are little footbridges, some are part of lock gates on canals, some bridges cross roads not not rivers, some are parts of lock gates, some are places where the river briefly disappears into a pipe to cross under a road, some are motorway bridges or train bridges that have no pedestrian walkways, some only carry and some only carry pipes or cables across the water. In the heart of Bristol even the platforms of Temple Meads railway station cross a river, though you wouldn’t be able to step off the platform on the other side. Which of these counts as a bridge? Similarly we can ask what is Bristol? The extent of the city? The historical city wall? The county? or maybe the church parish, or the election district?

For about two weeks I was stuck on these problems, of course I could have made arbitrary choices, but I wanted a principled solution. Eventually an idea occurred to me: Consider the original problem from Königsberg. Original challenge was about finding a nice walk that the gentry of the city to enjoy. So everything that was not legally walkable was out. Furthermore the Königsberg Bridge Problem is a puzzle. Clearly including a bridge across a road would not add to this puzzle as one can quite easily walk around these bridges if we needed to avoid using them a second time. For the same reason all bridges across minor canals etc that are disconnected from the main river do not add to the puzzle and should therefore not be considered.

Regarding the boundary of Bristol, I chose the county boundary as other potential areas cut some bridges in half. Interestingly that left me with one edge case as there is a motorway bridge that crosses the airspace of Bristol County while starting and ending elsewhere—I decided that it does not count for the challenge.

Clifton Suspension Bridge

Eulerian Ramblings

In 1736 Leonard Euler proved that a walk that crosses every bridge exactly once is possible if there are at most two landmasses that connect to an odd number of bridges. If no landmass has an odd number of bridges then the walk is not only possible but also leads the walker back to the starting point when they are done.

In Historical Königsberg seven bridges connected the two banks of the river Pregel and two river islets. The bridges were positioned such that one of the islets connected to five of the bridges, while the other islet and both banks had access to three bridges each. That makes four land masses with an odd number of bridges so now walk was possible.

Like Königsberg Bristol is build on the banks of a major river, the Avon, and also occupies two river islands, plus another very tiny one. But when I first started to work on the bridgewalk, there were forty-two bridges only one islet and one of the river banks connected to an odd number of bridges.

At the time of writing three more bridges have been built, so the number has increased to forty-five and now all the islands have an even number of bridges, which makes it possible to go on a walk that crosses each bridge exactly once and returns you to the starting point.

A First Attempt

Once I realized that a walk is possible I knew I had to try to walk it. If a walk is possible with a given set of islands and bridges there are typically many different eulerian walks. Finding a simple formula that can be used to compute the number of eulerian walks in a given network is an important open problem.

If you solve the problem on a piece of paper, any solution will do. But if you actually have to walk it, it changes your perspective: You don’t just want any bridgewalk, you want the best one. The best walk should be reasonably short but that’s not all, you want to stick to nice paths and avoid major roads. At the time I already done several exploratory trips to find out if certain bridges are walkable, and those gave me a good idea on where I wanted to go, and which parts I wanted to avoid.

Mathematics has relatively little to say about optimizing your bridgewalk, which is surprising given that the Königsberg Bridge Problem is very well studied in many other ways. So I developed my own method for the optimization, which worked well.

My first attempt at the walk ended in disaster: On the way I passed the Temple Meads Relief Line Bridge, an enormous steel girder railway bridge, painted in bold blue. In aerial photographs it had looked quite narrow, only carrying a railway track, so it wouldn’t count for the puzzle. But now as I walked past it I saw stairs leading up to it, and indeed what was once a railway bridge had been converted to a pedestrian crossing, so it did count! With the addition of this bridge the walk was still possible, but I had started in the wrong place.

Clifton Suspension Bridge seen from below

The Second Attempt

I took two weeks to discover from this first failure before I made another attempt. But I had gained valuable information; on the first try my progress along the sequence of bridges had been slower than expected. On my second attempt I got up up shortly after 4am on a grey Saturday morning. I got ready, made two giant sandwiches, packed a lot of water, and then walked about half an hour to the starting point of the actual bridgewalk.

I had made a plan, shown above, that accompanied me on the walk. But it showed only the sequence of bridges. I had planned for some shortcuts I wanted to take using the excellent maps of the British Ordinance Survey. But mostly I was finding the routes between bridges while I was walking. I recorded some video on the way and walked into many dead ends, including two very bad ones. All the way I was worried that I would find another unexpected bridge, but I didn’t.

The low point came before the last bridge, that was 48km or so into the walk. I had been walking back toward the centre of Bristol along the Avon, on a beautiful, serene trail. But now night was falling, my last sandwich had long been eaten and now I had to climb up 75 metres to the final bridge. After walking all day without a break it felt like I was climbing Everest. Fortunately this was a part of the walk that I knew well, the trail up to the bridge went through Leigh Woods, one of the wilder and more magical parts of the countryside. In the evening light it was beautiful, but I also knew that within the next half our it would become pitch black, and I did not fancy getting lost in the woods at night.

I made it to the top of the woods as the light slowly faded. From there it was a short walk along a road. Shortly after 6pm I crossed Clifton Suspension Bridge, completing the walk. I had walked all day and covered 33 miles (53 km) but I had crossed each bridge exactly once.

Reception

Up to this point I had thought of the bridgewalk as something that I did for myself. But while walking I had some time to think, and I thought if this is interesting to me maybe it is interesting for other people as well—perhaps I should write a press release? Normally we do press releases for scientific results, but this was interesting, wasn’t it, maybe someone would pick it up?

Next thing I knew the bridgewalk was getting double-page coverage in major newspapers and I was talking to the BBC. Even before that my scientific work had received coverage, arguably even in more important media such as the New York Times, Wall Street Journal or National Geographic. But his was different, particularly the detailed fairly high-intensity coverage meant that for a week or so people recognized me in the street.

Also for he first time in my life public attention somehow sustained itself. I received a lot of requests for walking instructions. Other people did the walk and posted on social media about it. A cancer survivor ran the route as an extended marathon to raise money for a cancer charity and that brought a new wave of attention. People started to blog about it, and I received more requests for interviews. Many people did the walk for fun or as part of smaller fundraising events. Later it would become the headline event of Bristol Giving Day in 2022 and Christian Climate Action declared it as a pilgrimage.

Alan Champneys asked me if I wanted to contribute a chapter to the book “50 Visions of Mathematics” that was being published for the fifties anniversery of the UK’s Institute of Mathematics and it’s Applications (IMA). I agreed, wrote my chapter and only later discovered that other contributions were by famous authors such as Simon Singh.

During an interview for Better Bristol, I met Jeff Lucas, the president of Bristol’s Civic Society. Jeff became fascinated with the idea of the bridgewalk. He started researching the bridges and the stories they tell, and started taking pictures of all the bridges. Jeff is an amazing photographer, and he aimed to photograph every one of the bridges in ways they hadn’t been shown before. Eventually he compiled his photographs and stories about the bridges into the book “From Bryckstow to Bristol in 45 bridges,” to which I contributed some mathematical explanations.
The Bridgewalk Today

Today the bridgewalk remains a fascinating walk, weaving back and forth around the Avon and its tributaries. Clifton Suspension Bridge remains the highlight of this walk, but the each bridge has a story to tell, and each has its own character, from the steam age aestetics of Vauxhall bridge, to the playful Victorian Gaol Ferry Bridge, the brick arches of Portway Trym Bridge, the ultra-modern stainless Meads Reach bridge, or the mile-long brutalist Avonmouth bridge.

With the addition of more bridges the bridgewalk is now a cyclical route.

As different as the bridges are the parts of Bristol that the walk visits, form the lively city centre to the serene Avongorge, the Street Art of Spike Island, the Villa’s of Clifton and Victorian factories on Silverthorne road. The walk has it all.

Today the Bristol Bridge Walk has a wikipedia page and a lively community on facebook, which has recently also become a centre a rallying point for activism to keep Bristols bridges and footpaths open to the public. The facebook community has current information on the bridges and many people posts the photos of the walk there.

There are now also alternative routes for bikes and mobility scooters.
Some Links

Bridgewalk on Wikipedia

Walking Instructions and Map @ Bristol Civic Society

Facebook Community

The Book: “From Bryggstow to Bristol in 45 Bridges”

If you complete the walk you can get the official Bridges Walk pin for this and other enquiries please contact: bristolbridgeswalk@gmail.com
2026-08-13
Königsberg Bridge Problem

2026-08-09

Königsberg Bridge Problem

Exploration, Networks, Research, Try-it-yourself

The foundations of network science is graph theory, and at the beginning of graph theory stands one problem that started it all: The Königsberg Bridge Problem. The problem poses a simple challenge:

“Can you go on a walk that crosses every one of the seven bridges of historical Königsberg exactly once?”

No, you can’t. Some of the historical bridges in Königsberg don’t even exist anymore. But could you have done it, if you had lived in historical Königsberg? Could you do it in your city today? Thinking about this leads down a fascinating rabbit hole.

Eulerian Walks

The map of Königsberg shown above was published in 1652 and is commonly attributed to “Merian-Erben”. But Merian-Erben isn’t a person. Erben means “heirs” in German and Merian-Erben are the heirs of Matthäus Merian, a famous Swiss engraver working in Frankfurt, Germany. When Merian died in 1650, the map of Königsberg was found among his estate, and two years later his heirs collectively published it with the attribution “Merian-Erben.”

Historical Königsberg looks like a pretty town, doesn’t it. Wouldn’t it be nice to go for a little walk on a Sunday afternoon that crosses every bridge exactly once? But can you? It would take another eighty years till this puzzle was solved.

Today we say that a walk that crosses every bridge in a city exactly once is an eulerian walk. The name honours Leonard Euler who solved the Königsberg bridge problem in 1736 and in doing so created a new branch of mathematics.

Euler was well aware that the problem could be solved by carefully listing all possible walks and then checking each one in tern to see if it was a solution, he writes:

The particular problem of the seven bridges of Königsberg could be solved by carefully tabulating all possible paths, thereby ascertaining by inspection which of them, if any, met the requirement. This method of solution, however, is too tedious and too difficult because of the large number of possible combinations, and in other problems where many more bridges are involved it could not be used at all. When the analysis is undertaken in the manner just described it yields a great many details that are irrelevant to the problem; undoubtedly this is the reason the method is so onerous. Hence I discarded it and searched for another more restricted in its scope; namely, a method which would show only whether a journey satisfying the prescribed condition could in the first instance be discovered; such an approach, I believed, would be much simpler.

Leonard Euler, Solutio problematis ad geometriam situs pertinentis, 1736.

Even in 1736 it would have been possible to solve the problem in this way in an afternoon: With only seven bridges, there are only 2520 paths that need to be tabulated and checked. But Euler’s idea is much better. It hints at tow powerful principle: Simple models and Economy of Information.

Simple Models

Euler’s first insight was that everything becomes easier when we eliminate the needless information. Merian’s map is nice, but all the details it shows are not relevant to the problem, and they obscure the view of the bridges.

Problems become easier when we focus on the bare necessities needed to capture their essence. This is the reason why we make models, to simplify a world that is otherwise to complicated to understand.

Euler’s map of Königsberg, shown above, shows only the elements that are necessary to capture the essence of the problem: the land masses and the bridges connecting them.

Building on Euler’s work, there is an even easier way in which we can represent the structure of historical Königsberg. We represent the different land masses by the nodes of a network and we represent the the bridges connecting them as links between these nodes.

Economy of Information

The other principle at which Euler hints is Economy of Information. In a nutshell: If we pose a question for which a typical answer would contain a lot of information, the answer will often be difficult to find. We need to pay a price for the information that we demand. A longer answer will take more work to find. Conversely if we ask a question that can be answered more quickly, then there is often an easy way to find the answer.

If we ask how to walk across every bridge exactly once, the answer we are asking for is the walk itself, e.g. a set of instructions containing the sequence of bridges that we cross. By contrast if we ask is a walk possible at all, the answer is a simple yes or no.

Euler focussed on the question why a walk that crosses every bridge exactly once might not exist. To discuss failure we need the concept of node degree. In our network representation of the city, each network node represents a landmass and each link represents a bridge. In network science the degree of a node is the number of links connecting , so in our case the degree of each landmass is the number of bridges connecting to it.

Solving the puzzle

Now consider what happens if our network contains a node of degree-one, a landmass with only one bridge connecting to it. If we don’t start the walk in this node of degree one, then we can only reach it by crossing the one bridge that connects to it. Once we have done that there is no way to leave the node again without using the bridge another time, which means the walk must end there. In summary that means: If a node of degree-one exists in the network, then it is either the start or the end point of an Eulerian walk or no such walk exists.

By contrast a node of degree two does not have such a negative impact on our walk. With its two bridges we can use one to arrive and one to depart, and hence we cannot trap ourselves there. Only if we start our walk in a node of degree two, the walk will end when we return to it.

So what about nodes of degree three? If a landmass has three bridges connecting to it we must visit it at least twice to use all bridges. On our first visit we arrive on one bridge and depart on another, without problem. But we have to visit another time to use the last bridge, and now this last bridge is the only one left—what started out as a node of degree three has now effectively become a node of degree one, so on our second visit we will get trapped in this node.

By the same reasoning all nodes of odd degree behave like nodes of degree one. They are either the starting point of the eulerian walk or the end point or no eulerian walk exists. And since each walk can only have one starting point and one endpoint, no eulerian walk can exists in networks that contain more than two nodes of odd degree. If a network contains exactly two nodes of odd degree an eulerian walk exists. In that case we can start the walk in either of the two nodes of odd degree and then the other will be the end point. Finally, if a network contains no nodes of odd degree we can start the walk anywhere and the eulerian walk will be a circuit that eventually ends back at the starting point.

So what about Königsberg?

The Truth about Königsberg

In the the Königsberg network one node has degree five and three nodes have degree three, so we have four nodes of odd degree. This proves that that no eulerian walk exists—we cannot go on a walk that crosses each bridge exactly once.

Today the bridge problem is often portrayed as a serious problem that people wanted an answer for, but this is not what it was. In the eighteenth century their was no internet, no youtube, no social media. Instead people turned to books and wrote letters. Many, many letters. Even Kings and Queens of the time were keeping up with recent developments in science and mathematics, and members of the high society would challenge each other to solve puzzles. Occasionally they might send someone an insolvable puzzle as a joke. The most notorious one of these joke puzzles was the Königsberg bridge problem—a form of 18th century trolling.

A good troll problem must not only be unsolvable, but it must also seem simple to ensnare its victims. Indeed, the Königsberg bridge network, is the smallest, simplest network for which there is no solution without this being immediately obvious. To be unsolvable a network must have at least four nodes, each of them with an odd degree. We don’t want any node to have degree one, because that put people immediately on the right track to the solution. So far the simplest network that meets these criteria, would have four nodes of degree three. But there is only one such network, and it forms a tetrahedron, a network that is so symmetric that we by exploiting the symmetry we can reduce the number of walks that need to be tried to just two—clearly not good for trolling. The next simplest unsolvable network has three nodes of degree three and one node of degree five and it is the Königsberg bridge graph.

Euler would not be trolled by it. Instead he proved that the bridge problem was unsolvable in such an elegant way that everybody could understand the proof. Because the problem was already notorious the news of the proof spread like wildfire throughout Europe. Essentially by thinking about the city as a system of nodes and links Euler had invented a new way of describing the world with mathematics—a new way of modelling. Euler said he had wanted a proof that mirrored the elegance of geometry, and he called his new way of thinking Geometria Situs, the geometry of position. Later J. J. Sylvester coined another name for this kind of mathematics: Graph Theory, and today many would call it Network Science.

Bristol Bridges Walk

In Königsberg no Eulerian walk exists. But the city of Bristol a walk is possible. I walked it and then everything went a little crazy.

You can read the full story in this post.

Try It Yourself

Why not try it yourself. You could design a bridge walk for your own town and perhaps it will become as popular as the one in Bristol.

2026-08-09
Patterns of Life

2026-08-07

Patterns of Life

Art, Maths, Research, Simulation

Life is full of patterns: From the stripes of the tiger to the spots on the leopard, the coats of animals come in many colours and shapes. Still there is a shared element between all the biological patterns, a similar aesthetic. Some patterns are clearly shaped by morphology, e.g. the wings on a bird having a different colour from the rest of the body. But there are also patterns that exist on otherwise uniform parts of the body, such as the stripes of the zebras and tigers.

We did a mathematical study of a mathematical instability that creates organic spot and stripe patterns, and even rotating spiral waves. The result was a highly cited paper and interactive exhibit in a major science museum.

Stripes on a Tiger

In 1952 Alan Turing proposed that patterns can form as the result of diffusion. What a crazy idea! In our daily life we know diffusion as the great equalizer. Clearly the reason why the tiger has stripes is rooted in the differentiation of hair cells, some of which become black-hair-producing cells, while others become orangy-yellow-hair-producing cells. Because the different colours are organized in neat stripes the hair cells must somehow coordinate. At some point of deveolpement of the tiger the hair cells must have had a sort of discussion on who is going to produce what kind of hair.

The hair cells themselves are too big to move around much, so there dialogue must be facilitated by much smaller signalling molecules that carry information between the cells. The precise mechanisms are still unknown, but for the sake of argument we can assume that there is one type of signalling molecule that signals blackness while another signals orangeness. When the time arrives for a hair cell to make a choice it looks at the signalling molecules that arrive at its cell membrane. If the majority signals orange, we get an orange hair, otherwise we get a black one.

In our experience diffusion tends to average out concentrations. Even if we started out with more signalling molecules for, say, black at one spot, we would expect that over time diffusion would lead to an equal distribution of blackness over the whole tiger. Instead we get clearly defined stripes.

Turing realized that diffusion can not only wash out patterns, it can also produce them. This is possible when two different substances diffuse that react with each other either directly or indirectly, e.g. via the hair cell. In that case we can start with a blank slate where the signalling molecules are equally distributed and then watch them redistribute to form patterns.

A Puzzle in Marine Sediments

Many years after Turings foundational work Martin Baurmann, Ulrike Feudel and I became interested in pattern formation in marine sediments. Even tidal flats that look like uniform expanses of mud to the naked eye have chemical patterns. Imagine standing on a nearly flat plane of mud that is dozens of miles wide, but if you measure the chemicals in the sediments beneath your feet, the results you get in one spot are very different from the results you get only a hand’s breadth away. How is this possible? What causes these differences? Was it possible that the bacteria in marine sediments coordinated in a similar way as the hair cells in the tiger?

We made a simple mathematical model, which is still perhaps the simplest ecological model that exhibits spontaneous pattern formation. It showed that chemical pattern formation in marine sediments was plausible. More importantly the model was so simple that we could study the pattern-formation mathematically, but there was a surprise waiting for us.

When we change parameters of a system gradually, the behaviour of a system typically responds gradually as well. But there are specific parameter thresholds at which the behaviour changes radically. These thresholds are called bifurcations and form the boundary between qualitatively different types of behaviour. The onset of pattern formation is such a bifurcation, which is called Turing bifurcation, in honour of Alan Turing. But when we analyzed our mathematical we found not only the Turing bifurcation, but also a Hopf bifurcation which marks the onset of sustained oscillations in the system.

Click here to open the paper

This Christmas card of the Max-Planck Society uses a pattern from the project

More Patterns and a Museum

The Turing and Hopf Bifurcations are two different fundamental instabilities of spatial systems. In our model these instabilities intersect and interact, and if parameters are chosen close to the intersection then pattern formation goes wild. Some first simulation runs revealed a few stationary patterns but many more showed complex spatiotemporal structures that combined oscillations with spatial shapes. Clearly we needed to do many more simulations to explore what else was possible in this system.

Now there was an idea: We needed to run a lot of simulations of fantastic shifting patterns that created themselves out of nowhere, sometimes collapsed back to nothing, sometimes settled into predictable patterns and sometimes descended into chaos—it was fascinating to watch.

At the same time a new science museum had just opened in Bremen, the Universum Science Center. In a conversation it came up that science museums only ever show the results of science, but not how the science really happens, and we had the perfect solution: Our simulations were science being done, and they were interesting to watch.

We approached the museum and obtained some funding from the local EWE foundation. The money paid for a fancy exhibit that we called “patterns of life,” which contained four computers and four screens that were constantly running simulations. Some interactivity allowed visitors to identify interesting parameter values for more simulation runs and thus become participants in the research.

The “patterns of life” exhibit ran for four years in the permanent exhibition of the Universum Science Center, afterwards the installation had a second life in the lobby of the Institute for Chemistry and Biology of the Marine environment (ICBM).

2026-08-07
Creative Reactions

2026-08-05

Creative Reactions

Art, Drawing, Maths, Research

Living in Bristol was fun because there is always something new and exciting happening in the city. One day I became aware of the creative reactions program, an initiative to foster contacts between art and science. Through this program I got into contact with graphical artist Frankie Partridge, we spent long hours talking about complexity and eventually created some art together.

Cohesion and Chaos

Our artwork Cohesion and Chaos captures the attempts to rein in the complexities of life through research. The right-hand-side of the artwork depict variations in the style of figures from historical scientific manuscripts. The left-hand-side is a two-dimensional bifurcation diagram of the Takens-Bogdanove bifurcation combined with various phase portraits—one of the transitions that can occur on the route from regularity to chaos.

Reception

Cohesion and Chaos was shown at Hamilton House Gallery in Bristol, UK, for two weeks in 2016. Later that year it was also exhibited as part of the North Bristol Art trail.

The initial run of fourty limited edition prints has sold out.

2026-08-05