Don Quixote tilts at Zeta functions

October 11, 2011

A friend of mine, Rohit Gupta (@fadesingh) has been doing some of the most creative mathematics communication out there. Using myths, stories, puzzles and poetry he has been making deep questions of mathematics accessible to others in online workshops and now a newspaper column in India. He is about to start a crazy and fascinating project, taking on possibly the greatest challenge of modern mathematics, the Riemann Hypothesis, which plunges the depths of the mysterious structure of the prime numbers. This would be an ambitious project for a group of mathematicians to take on. Current wisdom is probably that there are not even realistic routes to solve the problem. For a group with little or no mathematical training it is just crazy. That is what I love about the project.

Regular readers of this blog will know that I am a fan of impossible quests, and this one comes damn close. In the classic tradition of Don Quixote (which even mentions prime numbers) the value of a quest lies in the seeking, not the goal. Why should the privilege of failing to prove the Riemann Hypothesis be reserved to mathematicians? At worst a group of people will learn a lot, just getting an idea of the prime number theorem, or a clue about what a zeta function even is, is already an achievement. Can that be bad?

Even better the project emphasises that whatever your philosophy of mathematics, its actual study is a very human process. Baring some fairly extreme situations, if the Riemann Hypothesis is proved it will be by humans. Possible special and weird humans, but humans nonetheless. Just as the problem itself was discovered/dreamt up/found by one. Changing the perception of mathematicians as priests with almost magical abilities, to smart, professionals who have been through a tough training again cannot be a bad thing.

I am therefore proud to be part of this project, and see my role as that of Sancho Panza, sometimes bringing the flights of fancy down to earth, but increasingly fascinated and invested in the quest and where it might lead.

Of course it doesn’t hurt that the first stage of the project is playing with quasi-crystals, which have been a large part of my research life.


Magnetic Klein Quartic

October 2, 2011

The Klein Quartic is a absolutely fascinating object and worthy of a post in its own right, or even a book. It is clear evidence of the explosion of imagination and creativity in geometry that was taking place in the nineteenth century, as it cut its ties to the “real world”. Since then it has turned up all over mathematics.

One way to consider the Klein quartic is as a generalisation of a regular polyhedron. The tetrahedron has three equilateral triangles meeting at each corner, the cube has three squares and the dodecahedron three pentagons. Three hexagons gives a tiling of the plane. Why stop there? What about three regular heptagons? There are important reasons why this does not work in a simple manner. By playing fast and loose with what we mean by “regular heptagon” however we can do something. One object we can make is the Klein quartic. It does not produce something like a sphere, as the tetrahedron, cube and dodecahedron do, instead it is more like a pretzel with three holes.

Combining these ideas with little spherical magnets, we can make a model of the Klein Quartic. To do this we obviously have to start by making a heptagon

You start with a ring of seven balls, then put another ring of 14 balls around it. Note as this happens the heptagon buckles into a saddle shape. This is because the balls naturally create angles of 120˚ at the corners. As we move round the shape therefore we turn through a total of 7*120 = 480˚, this is greater than 360˚. We say the resulting surface has negative Gaussian curvature. We may also consider the length of the second loop. It is roughly distance 2 from the centre of our shape, yet it has length 14. If it were a circle of radius 2 the circumference would be 2*2π, which is less than 14.

Two of these heptagons can fit together on an edge:

For fans of Indra’s pearls and sphere reflections the balls make a pretty pattern.


As the angle at the corner is 120˚ three will fit round a corner:

We could now continue this, bringing three heptagons together at each corner, but we want to create the finite object. Next attach an additional heptagon to each of the outer three:

Now connect the three outside heptagons together. to make a surface with three holes:

You need to repeat this four times, using a total of 24 heptagons. As you make them, be careful of one thing, the magnets line up so that you get  all N poles on one side of the surface and all S on the other. As you connect each surface, therefore, make sure that it agrees with the others:

When you have all four, put one at the center and then connect the others to each of its four holes

To finish, technically we should connect up the remaining six holes so each branch is connected to both the others. The resulting shape has three heptagons meeting at every corner, and a wonderful collection of symmetries many of which cannot be easily seen in this model, or any model in 3d!

Just for kicks, lets finish with the work of one of Klein’s contemporaries a Möbius strip:


CAMel

September 29, 2011

CAMel is a project to develop Rhino Grasshopper components for CAM (Computer Aided Manufacturing). Hence the silly name. It is very much work in progress, but if you are brave enough, here is a first release. All images and the video on this page are of a machine running GCode generated by CAMel.

Download CAMel 0.12
Download Rhino file (only needed if you want to see the example setup).

At present the components are just clusters with scripted components written within Grasshopper. The next major step will be to convert this into a proper grasshopper plug-in. This release has a grasshopper component with some documentation (there is a little more inside the clusters). All the code is CC-BY-SA licensed, and of course it should be noted that this is very much “use at your own risk”! My belief is that Grasshopper provides a natural environment to experiment with creating your own toolpaths. The purpose of CAMel is to make this process as easy as possible by giving the tools to convert simple toolpath ideas into usable paths and then exporting the GCode that will drive a machine.

The main components are as follows:

  • GCode Writer: Converts lists of points, vectors and feed rates into GCode for the machine.
  • GCode Checker: Reads GCode and checks and optimises it. For example a 5-axis machine can usually obtain any tool angle in two different ways. This selects the better angle. It will also give warnings of undesirable behavior in the GCode.
  • Surfacing: Creates a toolpath to cut an arbitrary surface (very rough version, designed to test others)
  • Swarf cutting: Creates toolpath from information about the movement of the tip of the tool and the point in which the tool enters the surface. For a 5-axis machine these paths can be quite different.

The code is currently set up for a single machine, I am happy to try to help adapt it to other machines (other commitments allowing) so get in touch if you are interested.

These components and code were developed with Santiago R Perez 21st Century Chair of Integrated Practice at the Fay Jones School of Architecture, University of Arkansas. I work in the Mathematics Department at the same university.

Earlier experiments with swarf cutting.


The University Project: Stories and Science

September 25, 2011

For background you might start with these two pieces from Dougald Hine:
About this university…
The University Project: Five Reasons

I can’t tell stories. I am a mathematician, I find rules. I want to break everything down into its simplest components, to things that feel self-evident. This can be stereotyped as reductionism, even criticised for taking away the beauty and mystery of the world. That is the cultural wars crap. To me the process of seeking the simple, understanding the rules, is itself simple. To cut away the things that are simple so that we can get to the true complexity. To see problems clearer by cutting away the simple stuff that initially looks complicated. Right now my rules based understanding is screaming:

The world needs story tellers and stories

Stories that et us work more effectively with he world. Stories that allow us to grab the understanding gleaned from some deep scientific study without having to get to the bottom of its details.

Think about fairy tales, there is something magical about them and it is not just the witches. On the surface they are fantasy, things that clearly made up, 1000 year sleeps, pumpkins becoming coaches, houses made of cake… Yet at their core they have deep psychological truth and wisdom. They can help prepare children for the darkness of the world around them. In fact any work of fiction is by definition lies, but novels have had as great an effect on my life than just about any understanding I got from science and maths.

You do not get this wisdom just by being a story. It comes from the authors wisdom, understand and beliefs about the world. And there’s the rub. The understanding of the world that we have been able to grasp with the tools of mathematics and science is immense and detailed. Even the experts can only grasp parts of it. In fact the amazing work of Gödel, Turing, Church, Post, Chaitin and others means that this can be exactly quantified. To butcher another great story teller:

And therefore as a stranger give it welcome.
There are more things in arithmetic, Horatio,
Than are dreamt of in your philosophy.

Yes, humble arithmetic can be studied for as long as you care and yet still reveal new secrets. When this is the case, how do we expect someone who also needs to learn the delicate arts of telling stories, or performing to also gain a deep understanding of science or medicine. Just as we cannot expect those who have spent many years learning how to create science to also tell compelling stories. There have been many examples of course, but very clever people in a single field are rare, so we should not just wait for the few who are brilliant in more than one, when we can bring the one-fielders together.

We need the skill and art of storytellers forged by the ability of science to cut through the crap and give a sense of what is real. Stories that on the surface are engaging fictions but whose heart and core message can be backed up by spreadsheats and data.

This bring me to the university project. Yet when I try to say why it becomes difficult. Everything above is really just part of the classical ideal for the university. Universities still act as the haven for many, many wonderful things. Why then do I feel we need to explore alternatives? Yet I do. I feel that due to a combination of pressures universities do not nurture such collaborations in the ways they need to be. The value system within academia has become too focussed on ability within a specialisation and a certain value system based on certain forms of publication. This is especially true at the hiring level. Working outside a discipline is consider a great thing, but only after you have mastered your field. The problem to me is that working outside an expertise is not a trivial task. It needs to be studied, worked on and developed. Successful collaborations can take years to develop real output, and there are other collaborations that last years without getting there.

The university project feels different, placing open connections at its institutional core. Using play and friendship as ways to overcome the barriers of the jargons and fixed ideas we have to develop to become successful specialists. It is not about replacing the university, or even reinventing it, but about giving options opening new paths to the beautiful concept:

The cultivation of knowledge.


These ideas have been fermenting for a long time, and have developed during my involvement with the University project and the discussions and ideas around it, most notably the thinking of Dougald Hine (of course) Alex Fradera, Nick Stewart and Rhett Gayle. These crystallised into a concrete idea when I read a tweet from Vinay Gupta.

I am not claiming any particular originality in the thinking, and there are many examples of the sort of combination that I ask for. I want this to increase, not to start! As examples you can look to authors such as Borges, Nabakov, Perec, and Pratchett, and coming from the science side to tell stories Asimov and Sagan. It does feel that this sort of endevour is growing. The theatre production A disappearing number by Complicté working with mathematician Marcus du Sautoy is an incredible example. It is discussed in the broader setting by Joe Winston, whose work on stories, beauty and education is wonderful.

Other people telling stories with and of mathematics include my friends Paul Prudence and Rohit Gupta. Both of whom provide lyrical beauty and whimsy to mathematical ideas.

There is so much possibility  and to me the grand push of the university project is to try to explore and find ways of unleashing it to the true benefit of humanity and the world.


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