I have been thinking quite a bit recently about ideas of knotting and weaving. There will probably be another post on the theme soon. As a mathematician it brought me straight back to Knot theory, I love looking at the strange images that appear on the blackboards in the lectures and offices of topolgists, many of which contain knots. This video lecture from Elvis Zap is a classic example (even if you cannot follow, just sit back and enjoy the drawing!). Not to forget the beautiful uses knotted designs have been put to outside of mathematics.
At some point during this I needed something made out of metal, and decided to bend some wire into a trefoil. It was satisfying, so I though I would look online to see if I could find collections of physically made knots. These were surprsingly hard to find. Therewereplentyofexamplestobefound (even the Museum of Mathematics‘s famous knotted bagel), but I could not find any systematic collections. So I decided to make my own, using this knot zoo for reference. Here are the knots that can be drawn with seven crossings or less, using Conway’s tangle notation:
It was great fun making the knots and I encourage anyone who studies them, even idly, to have a go. I felt the knots themselves come alive in my head as I made them. I started to think how the knots could be put together out of sections of twists, further study of this lead me to tangles and Conway’s notation. You might notice that this came late as the written labels on the knots are the more commonly used Alexander–Briggs notation. That is a lot less satisfying as after the number of crossings the numbers to not refer to the properties of the knots.
In addition making sure that the wire holds naturally in shape without touching itself is great for 3d intuition. One thing that struck me as I started to bend the wire was how 2 dimensional most knot images are. The crossing number is a classic example of this as it is a 2d not a 3d property. There are, of course, good reasons for this both in design and exposition, but it was interesting feeling how the knots changed as you allow to move more freely. Of course this had some issues when I came to present the knots here, of course in 2d (I hope I managed to get all the pictures so spare crossings are easy to remove!). A video might work slightly better:
I have a weird collection of skills. Mathematics, talking about mathematics, art, making…
I am certainly missing opportunities, maybe because few know the skill set even exists! So its time to advertise myself. Perhaps you are looking for someone who can…
Do mathematics at a research level, especially:
Geometry, understanding the spaces we live in and more exotic ones.
Tilings and patterns.
History and culture of Mathematics
Talk maths in public.
Teach (and be creative at it)
Program
Use computer manufacturing tools, Laser cutters, 3d printers, 3/5/n-axis routers.
Make Art and do Design
You need more evidence? I guess that makes sense. More details are below. If you still need to know more get in touch. I can provide references! (edmund.harriss at mathematicians.org.uk)
More details and evidence…
Mathematics: The heart of what I do, I have been an academic mathematician since getting my PhD from Imperial College in 2004. I have written papers, and been invited far and wide to talk about my work. See my CV for the gory details.
Geometry, Tilings and Patterns: I have a very strong understanding of the space we live in (and more exotic spaces). As this is a mathematical understanding I also have the tools to make this concrete, putting it into the equations and other things that computers can play with. My mathematical research has looked at tilings and patterns. Especially substitution tilings a sort of scaling symmetry, I probably know as much about the Penrose tiling than anyone else alive or dead!
Teaching: I want to teach people to actually think mathematically, not just get the rules that can be followed to a right answer, and have had success with it. Of course I can teach a traditional maths course and these are often necessary to get the bulk of material across, however I have also worked with more innovative courses. That is why I came to Arkansas. I wanted to teach MATH 2033 the conspiracy or mathematics course designed to corrupt people into the subject by giving a glimpse of undecidability, game theory, 4 dimensional geometry, hyperbolic geometry, topology, codes, sphere packings… The students then have to come up with their own projects and, as could be expected often get incredibly creative.
Making: I make things, normally focussing on explaining mathematics. I even have my own Laser cutter! I designed some larger versions of the hexayurt, a simple building made, without waste from 12 sheets of plywood or other materials. I am currently working with the FabLab at the architecture school here at the University of Arkansas, and am writing software to drive their 5-axis router.
Continuing the theme of maths sculptures interacting with snow fall, here are some pictures of my bamboo star. The original design was found by Akio Hizume, and I was introduced to the idea by Chaim Goodman-Strauss. The design takes 30 lengths of bamboo, arranged in fives. Each group of 5 pass through two opposite faces of a dodecahedron, as a pentagon rotated slight with respect to the pentagons they pass through. The 30 lengths weave together in the middle, needing no other support. Though over time the star does tend to sag!
Similar designs can be achieved from the other regular polyhedra. Can you work out the polyhedron that corresponds to this pencil design?
Last semester I offered my students $1,000,000 dollars. They turned me down. This was lucky, despite the money and glamour of academic mathematics, I do not have a million dollars. The game was simple. The class of 100 each had to write a number. The highest number won. Of course there was a catch, the prize was $1,000,000 divided by the winning number. The best outcome for the students as a whole would come if everyone wrote 1, $10,000 is not a bad return for a lecture. Of course if everyone is writing 1, the person who writes 2 wins and makes far more for themselves. What happened?
I did tell the students that they should all cooperate and write 1, explaining how this was the best outcome. Some very trusting students actually wrote 2. This was actually rather sweet, although they were out to win more for themselves they felt that everyone else would be looking out for the group. There were also more cynical souls, realising that they were not the only one they simply wrote the largest number that they could. As a result I did not have to pay out a single cent. I was slightly sad not to receive the answer “highest number written plus 1″, that others who ran the game have done. This gets even more interesting when two people do it!
Readers watching closely will recognise that this is a group version of the famous game prisoners dilemma. Another version was used in the final round of the UK TV series Goldenballs1. Watch these clips and try to guess what the people will do:
Having played the million dollar game, and watched the clips I asked the students what they would do. About 2/3 did say split, unfortunately for them only 1/3 of the students had written 1 for the previous game! This is not surprising, in the $1,000,000 game something was on the table. The high number writers still wanted the chance of winning the game, even if no money was involved. In a simple pole you get no benefit from admitting (even to yourself) that you would do over your neighbour.
Both these examples are compelling as they illustrate game theory in action. In the million dollar game the theory is actually being used to model the behaviour of a large group. A statistical study of the data from series of Goldenballs, reveals some subtleties. Even though they are playing the game just once over half of players actually did split. Tellingly, however, the average money in situations where both players split was lower than that on the table for stealers. Interestingly a higher proportion of people who used the word “promise” did split.
These examples can be studied using the mathematics of game theory, but they also reveal the problems, the exact pay off differs for each individual. It is not simply that it is hard to establish exact values, the values actually differ dramatically from one person to another. While it may be true, for example, that everyone has their price, the exact value of that price can dramatically change the game that is being played. Other factors (also varying from one individual to another) can also come into play. In a more far out example, in Goldenballs players might see themselves as playing primarily against the Television company. In this case part of the pay off would be seeing the company give out the money. This will definitely happen if they split, but might not if they steal. For these people the game changes from Prisoner’s dilemma to Chicken.
Does this mean that game theory is not worth studying, or even misleading? It certainly means that we have to treat it with caution. One of the founders of game theory Dr Strangelove John Von Neumann actually argued that it proved the necessity of the nuclear first strike during the cold war. Luckily for everyone his counsel was not followed!
Not quite John Von Neumann (though maybe based in part on him)
We associate mathematics with the unreasonably effective models we find in Physics, Chemistry and even Biology. In fact “mathematical” has almost become synonymous with precise. These models are certainly impressive, even beautiful, but game theory is not one of them. Game theory becomes powerful not as model, but as metaphor. It can help us understand what behaviours come out of situations with different payoffs. The lesson from prisoner’s dilemma is that people rationally following individual benefit in the society can lead to the group as a whole suffering. Historical events can also be analysed in terms of certain games. Although, unlike the models of physics the mathematics of game theory cannot be used to predict the future it can be used to understand the past and the present.
For more on the history and development of game theory and its potential social applications I can recommend these books:
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Technically this is not quite Prisoners dilemma, as, assuming your opponent is stealing there is no difference between splitting (you receive nothing) and stealing (you receive nothing).
Posted by gelada 
by William Poundstone
by Ken Binmore
