This is the Maths in the City walking tour of Oxford. We'll be looking at symmetry, geometry, and engineering using footprints, woks and marbles!
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 Oxford 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.
The tour stops are:
1. The roof of the Sheldonian Theatre
2. The Beehive
3. Frieze symmetry at the Ashmolean Museum
4. Labyrinth at St Giles’ Church
5. Mathematical Institute
You can download a printable copy of the tour.
1. The roof of the Sheldonian Theatre
The fascinating and inspired mathematics behind the construction of the Sheldonian Theatre allowed it to have the largest unsupported roof the world of the 17th century had ever seen.
When Sir Christopher Wren was asked to design the Sheldonian Theatre in the 1660s, he began with visions of the great amphitheatres of Ancient Rome. Oxford having rather more rain than Rome, his design of a modern amphitheatre was in need of a roof.
[Marcus du Sautoy at Sheldonian Theatre]
How would you use timber beams to build a simple roof?
The most obvious solution span the walls with the timber beams. But the dimensions of the Sheldonian are 70 foot by 80 foot and this simple roof design would need beams far longer than the timber beams available at the time. To overcome this Wren planned to use internal columns to support the roof but this was vetoed by the university officials – they didn’t want columns impeding any dancing at the venue! Today the Sheldonian is used for graduation ceremonies but back in the 17th century they had much wilder parties in mind!
Solving the puzzle of an unsupported roof
Wren couldn’t use internal columns to support the roof and he couldn’t span the space with individual beams. Therefore to complete the Sheldonian, Wren would need to build the largest unsupported roof the 17th Century had ever seen.
Luckily Wren had studied mathematics here at Oxford, taught by John Wallis. Wallis was the Savilian Professor of Geometry, a chair that still exists at Oxford today, currently held by Nigel Hitchens. It was Wallis' ingenious design that provided Wren with the answer he needed.
Wren's problem was that the walls could only support the roof at one end of the timbers. At first sight it might seem impossible to build a stable roof with beams that are only supported in one place – but if you have ever balanced on a seesaw then you’re already part of the way to the solution. Just like a seesaw balancing on its single pivot, Wallis’ ingenious idea hinged on statics: balancing all the forces involved in so they cancel out making the seesaw, or roof truss, stable.
If the timber beams are only supported at one end by the walls, how can you arrange them so that together they provide stable, strong roof? The answer is to interlock the beams.
Wallis’ devised an ingenious pattern of interlocking beams, so that every beam was supported at both ends – either by the walls or by other beams – while every beam also supported the ends of two other beams. So for every beam, the downward forces from those resting on it are balanced by the upward forces from the beams, or wall, supporting it. In an impressive feat of calculation, Wallis demonstrated that hisgeometrical flat floor could carry loads when supported by the walls alone by solving a set of 25x25 simultaneous equations using just pen and paper!
So Wren’s roof, inspired by Wallis’ design, not only keeps dancers (and graduates and their families) dry, it can also support significant loads. The Oxford University Press stored books on the first floor for many years, proving that you can build a strong stable roof supported by mathematics instead of columns.
• 3 flat sticks, about a metre long (the beams)
• 3 maths books
• Laminated picture of Wallis’ design
If you have a large tour group (over 15) you’ll need to divide them into two groups, with a set of 3 beams and 3 books for each group. Identify 3 volunteers in each group and place them in the centre of the group in a triangle roughly a beam length apart. These three volunteers will act as the walls supporting the roof, where the whole group will work together to design the roof.
Your groups can explore the principles at this site by trying to answer the questions in italics under the pictures in the explanation above:
How would you use timber beams to build a simple roof?
With the three volunteers standing about a beam’s length apart, ask them to build a simple roof. Mostly likely they will build one using single beams to span the walls. You can demonstrate how this design fails if the room is bigger than the length of the beams, by asking them to take a small step backwards so they are now more than a beam length apart.
If the timber beams are only supported at one end by the walls, how can you arrange them so that together they provide stable, strong roof?
The most successful approach with tour groups has been to just stand back and let the group get on with experimenting and solving this puzzle. If they are struggling after several minutes you could suggest they need to have the beams supporting each other, interweaving them in some way. The answer will be something like the following construction:
• End of first beam rests on one wall (you temporarily support other end)
• End of second beam rests on next wall and other end on middle of first beam
• End of third beam rests on last wall, other end on middle of second beam, and the centre of third beam supporting end of first beam.
To demonstrate that the structure is not only stable but can support significant weights, place several books in the centre of the arrangement (if the beams interlock close to the centre) or book at each of the three points where the beams cross. Encourage other people in the group to try holding the ends of the beams to feel the solidity of the arrangement.
2. The Beehive, St John's College
In St John’s College, Oxford, one of the buildings is hexagonal in shape. Was this hexagonal structure a whim of the architect? Why are most buildings square? What does all of this have to do with bees?
Students and busy bees
From children’s pictures to architect’s drawings, most of us imagine houses and buildings as being made up of squares and rectangles; not surprising given the majority of the buildings we see in the city around us are based on right angles. But we do have some fascinating examples of differently shaped buildings here in Oxford. Some are circular, such as the Radcliffe Camera or the Sackler Library, some have curved sides such as the front of the Sheldonian. But only one building in Oxford is hexagonal.
Making the most of your wall
So why should bees, or architects, use hexagons? For bees, making the wax for the walls of the honeycomb is a very expensive business. (A single bee produces just 1/12th of a teaspoon of honey in their entire lifetime. Bees in a hive need to consume 6-8 pounds of honey to produce 1 pound of wax, which means they collectively need to fly more than 6 times around the world to produce that amount of wax!) So understandably they would want to use this expensive resource most efficiently, building the largest cells possible for a given amount of wax.
If you know how much wall you have to use, say a fixed number of bricks or a fixed amount of honey, how should you build your room so that it encloses the largest space possible?
You can explore how to increase the area enclosed by a fixed length of wall using a loop of string and asking volunteers to add a corner to the room, one corner at a time.
You can explore this using a loop of string for your fixed length of wall. Starting with (an admittedly very useless) room with just two corners, each time you add in another corner (going from triangular, to square, to pentagonal, etc), you increase the area the string encloses. If you carry on adding corners you shape becomes more and more like a circle, and it is a circle which encloses the most area for fixed perimeter. (If you’ve got a mathematical bent why not try deducing the area of these shapes, and prove that it increases as the number of sides increases. You can check you calculations for the areas with those listed on Wikipedia).
Making the most of your space
A circle is the most efficient shape for a room on its own – it encloses the largest possible space for a given length of wall. Although we have some examples of circular buildings, including those here in Oxford, it isn’t a good choice for the shape of rooms within a building. [Why aren’t circular rooms common?] Circles don’t fit neatly together and wasted gaps of space would be left between circular rooms.
In order to make the most of your space you need a shape that tessellates or tiles the floor space of your building, just like the paving stones that neatly cover the courtyard near the Beehive.
There are only three regular shapes that can tile the plane – triangles, squares and hexagons. So by choosing hexagons, the architects of the beehive, both this building and that of the bee, have chosen the most efficient shape for a room – the hexagonal walls enclose the largest possible areas while wasting no space between them.
Bees have known about the benefits of hexagons for millennia, and we have suspected that hexagons were the most efficient way to divide up a flat plane from at least 300AD when Pappus of Alexandria posed this as a question. However this Honeycomb Conjecture was only proved mathematically just over a decade ago, by Thomas Hales in 1999.
Making the most of your energy
Although the architects of the Beehive were aware of all this maths, we can be pretty sure that the bees haven’t learnt any geometry. But nature, like mathematicians and architects, is keen on efficiency and will form shapes and arrangements that require the least amount of energy. The bees actually start by making roughly circular cells as these are the most effecient use of their wax. But as these pack together the walls bend to create a hexagonal arrangement. You can see how this happens by tossing a handful of marbles in a curved wok – the spherical marbles naturally settle into a hexagonal pattern.
Mathematically hexagonal rooms have many benefits, but they have some practical drawbacks too. Any of the students who have lived in these rooms will tell you that it can be quite hard to fit normal, predominantly right-angled, furniture into these hexagonal rooms. Some students apparently even went so far as to take to their beds with a saw to make them fit! Which is why most of us won’t be living in a beehive soon.
We've included questions (in italics in the description above) that you can ask the group to get them involved.
You can use a large loop of string (one about 6m long works welll) to explore how the area of a room (enclosed by a fixed length of wall) changes as you alter the shape. Ask for two volunteers and ask them to pull the loop taut., giving you a room you with just two corners – it’s not a very useful room as it encloses no room at all. The room becomes much more useful if you add a third corner, creating a triangular room which now encloses some space. Continue asking for new volunteers to add a corner to the room and note that the area enclosed continues to increase.
It is also useful to have a wok (or similarly curved dish) and a bag full of marbles. If you empty the marbles into the dish they will settle into a hexagonal pattern, demonstrating that making the honeycomb arrangement uses the least energy.
3. Frieze symmetries at the Ashmolean Museum
Artists have used friezes to decorate buildings for thousands of years. The symmetries of these patterns are key to their aesthetic beauty, and also to their mathematical significance.
In search of symmetry
Symmetry is something we all appreciate, whether in beautiful faces, stunning flowers or pleasing patterns. And we have been exploring symmetry for thousands of years, as can be seen in the carved stone balls produced over 3000 years ago in the neolithic period in Scotland. These are carefully decorated, most with evenly spaced carvings or knobs. And most remarkably more than half have 4, 6 or 8 symmetrically placed knobs, showing a striking similarity to the highly symmetric platonic solids that would be discovered a thousand years later by the Greeks.
One way symmetry has been explored over the centuries by artists from all cultures is through decorative patterns and friezes, such as those decorating the Ashmolean museum.
Magic tricks and foot prints
Frieze patterns, such as the simple beaded pattern around the columns of the Ashmolean, are strips decorated with a repeating pattern. As they are repeating patterns they all have translational symmetry – you can slide the pattern along until it matches itself exactly.
Symmetries are just like magic tricks. If you close your eyes and I perform one of these symmetry operations on the frieze pattern, it will appear unchanged when you open your eyes. This beaded pattern holds many other symmetries apart from translation. [What other symmetries can you see in this beaded pattern?] It would appear the same if I reflected the pattern horizontally or vertically, if I spun it through a half turn, or if I slide the pattern along and then reflected it horizontally (called a glide reflection).
The group of symmetries for this frieze pattern can be remembered by the simple name of "Spinning Jump". You can see the pattern of footprints left behind by spinning between jumps has the same symmetries as the beaded pattern around the column.
If you jump out this pattern and leave behind the cut-out footprints, it makes it easy to demonstrate that it has the same symmetries as the beaded frieze. Pick any of the symmetries, place four coloured cut-out footprints on the pattern, perform the symmetry transformation with these coloured footprints, and it will be clear that the footprint pattern will appear unchanged.
When symmetries fail
The spinning jump pattern has many symmetries but other frieze patterns, for example the "Sidle", are far less symmetrical.
What symmetries does this pattern have? Jump out the pattern, marking it with cut-out footprints. Use coloured cut-out prints to explore which symmetries hold, and which don't, for this pattern.
Unlike the Spinning Jump, which had all possible symmetries, the only symmetries which hold for the Sidle are translation and vertical reflection. Any other transformation will change the pattern. [Can you spot this frieze pattern on the Ashmolean?] This pattern can been seen in the egg and dart frieze running near the top of the building.
What other frieze patterns are there?
Some of the other frieze patterns you can see on the Ashmolean are the plait on the ceiling of the portico:
the plait running along the top of the walls:
and the spiralling pattern above the windows:
All three of these patterns have the same symmetries, a group of symmetries known as the "Spinning Hop".
What symmetries does this pattern have? Use the footprint pattern and coloured cut-outs to explore the symmetries.
Apart from translation, the only other symmetry that holds for these patterns is rotation: if you spin the pattern by 180 degrees it will appear unchanged.
The seven frieze groups
Mathematically we analyse frieze patterns by the combination of symmetries that can coexist within the frieze and how these symmetries interact with each other. You might think there are endless possibilities of different combinations of symmetries, but actually there are only 7 possible ways that these symmetries can occur together in frieze patterns. In addition to the Spinning Jump, Sidle and Spinning Hop groups we've already seen, there are four other frieze groups:
- the "Step"
- the "Spinning sidle"
- the "Hop"
- and the "Jump"
Artists have been playing with all seven of these patterns for thousands of years. But it was only in the nineteenth century that mathematicians were able to prove that no other frieze groups existed and every possible frieze pattern could be described by one of these seven frieze groups.
Mathematically symmetry is studied using group theory: each of the seven frieze groups above is an example of a mathematical group. One of the founders of group theory was the mathematician Everiste Galois. He made significant contributions to mathematics during his short life before he died in a duel in 1932. The night before the duel he passed the time writing letters and mathematical papers, which included him coining the word "group" for this type of mathematical object.
Group theory is used widely outside of mathematics too, in studying the symmetries of crystal structures and molecules in chemistry and in the Standard Model of particle physics currently being explored at the Large Hadron Collider.
We have only be able to spot three of the frieze groups decorating the Ashmolean. Can you spot anymore? And if you spot other examples of this beautiful piece of mathematics here, or in any city around the world, please let us know!
To make it easier for the group to see, hand around printouts of the image of the neolithic carved stone balls, the friezes of the Ashmolean and the footprint patterns for the seven frieze groups (you can download this footprint bingo).
We've included questions (in italics in the description above) that you can ask the group to get them involved.
To demonstrate the symmetries of these frieze patterns we jumped out the footprint patterns of the symmetry groups and marked the patterns with cut-outs. As indicated in the captions of the footprint patterns above, an easy way to explore the symmetries is to use extra coloured footprints that you can move to see if a transformation leaves the pattern unchanged.
4. St Giles' Church - Labyrinth
Labyrinths and mazes have fascinated people for centuries and appear many times throughout history.
The most famous example is probably that of the Cretan Labyrinth, home to the legendary Minotaur, a half-man half-bull killed by Theseus. We think the labyrinth looked a little bit like this.
Notice the labyrinth outside St Giles’ Church. This is similar to the Cretan Labyrinth. Labyrinths have come to mean a construction that takes you from a starting point to a destination by a torturous path. However, it doesn’t actually require you to make any decisions along the way! This labyrinth will take approximately 20-30 minutes to walk in its entirety, but once you’ve started you can just follow the path.
This is different to a maze, where decisions have to be made along the way about which direction to take in order to navigate the maze successfully. Most mazes these days are recreational, such as the mazes at Hampton Court Palace or Longleat. Solving a maze using mathematics is, however, an interesting problem.
Let’s suppose we know the map of a maze, and we are looking for a solution. We could do this by trial and error, of course, but as mathematicians we always like to find a reproducible and elegant solution.
Here is an example taken from PlusMaths Magazine – do look up the full article for more on this topic! The key to solving many mathematical problems is to transform a harder problem into one which we can solve more easily.
We want to get to the point M (and back out again).
We can transform a maze like this into a network by noting all the points where we have to make a decision, including to enter the maze and the destination. Then we can just draw a network.
Hopefully you can see that this makes it much easier to solve the problem. There is more than one route that can be followed. One is A > B > D > E > G > I > M. Can you find another?
A well-known example that uses this kind of technique is the map of the London Underground. Essentially we ignore the geographical locations or distances between stations in favour of showing connections, allowing users of the trains to navigate their way more easily around London – even if the route taken isn’t necessarily the shortest, we can usually identify the route with the fewest changes quite easily.
Why don’t you try doing this with the map of the Hampton Court Palace maze?
5. Penrose tiles at Andrew Wiles Building
No matter where you stand, the pattern in the pavement outside the Mathematical Institute never repeats. This is because it is a Penrose tiling, named after the mathematician Roger Penrose who invented it in the 1970s. Penrose tilings not only have many interesting mathematical properties, they also explain the structure of some unusual metallic crystals, called quasicrystals, that were discovered in the 1980s and won Dan Shechtman the Nobel Prize for Chemistry in 2011.
As you can see, there are just two types of tiles involved: two different rhombuses, with curved metal bands as an extra feature. You will recall that we saw lots of different types of symmetries at the Ashmolean museum. What is special about this type of tiling is that they are non-periodic, which means you can’t create the tiling by repeating any (however large) section. A set of tiles with this property is called aperiodic.
Penrose tilings not only have many interesting mathematical properties, they also explain the structure of some unusual metallic crystals, called quasicrystals, that were discovered in the 1980s and won Dan Shechtman the Nobel Prize for Chemistry in 2011.
This is the end of our guided tour, but while you are here, why not take the time to explore some of the mathematical features of the Maths Institute? The building has a number of features that you may find interesting. The guide has copies of these for you to make your own self-guided tour of the lower part of the building. Please do not enter any lecture theatres or classrooms.