• The collapse of Analysis
  The problems with infinity and infinitesimals had been known to the Greeks who discussed their paradoxes. In the seventeenth century, however, arguments and proofs involving infinitesimals became more acceptable. A powerful system started to emerge developed most clearly by Newton and Liebniz: the Calculus. Some still had issues, Berkeley famously mocked the “ghosts of departed quantities”, but most persisted and were greatly rewarded. The problems did eventually arise in mathematical settings, for example in Fourier’s study of heat, and it required most of the nineteenth century for Cauchy and Weierstrass to give a rigorous version based on limits.
• The collapse of Geometry
  With issues arising in the basic assumptions of Analysis, mathematicians looked for firmer ground. Many felt that Geometry and, in particular the great work of Euclid’s Elements might provide this. Unfortunately there was a wrinkle in the Elements, often known as the Parallel Postulate, a statement that seemed far too complicated to be an assumption. Many had tried to show that the parallel postulate could be proved from the other axioms, and failed. Some had actually glimpsed something further, notably Khayyam and Saccheri. In fact, as Bolyai and Lobachevsky would eventually show, there are perfectly good geometries that obey all the axioms apart from the Parallel postulate. With the zoo of examples that started to be considered, could geometry did not look like such a good foundation.
• The collapse of Arithmetic
  Instead of geometry, therefore other systems were considered, ideas from logic, and Cantor’s set theory were brought into play. Hilbert hoped that a system could be created that was consistent, complete and decidable. Many took on the challenge, Russell and Whitehead created the magnificent Principia Mathematica that famously does not prove that 1+1=2 until page 362. As the ideas started to become clearer, however, the way was left clear for a deeper issue. The results of Kurt Gödel showed that, if we want to include arithmetic, we cannot hope for a system for mathematics that is both complete and consistent. Furthermore,  we cannot even prove the consistency of our systems without resorting to a more powerful one (who’s own consistency cannot be proved without a more powerful system still). Some even wonder whether arithmetic is consistent!

As these collapses hit the ideas that fields rest on, one would expect there to be some consequences. Some areas that turn out to be fallacies. Yet this does not seem to be the case.  The fundamental ideas of calculus remained the same, although one had to be careful about the exact functions you were talking about. The discovery of non-Euclidean geometries simply revealed additional worlds, all the old results held but some now needed to note an additional assumption. Even the work on undecidability leads, most obviously through Alan Turing, to the theoretical underpinnings of computers. In fact studying the deeper issues seems to open up new areas but not harm those that have been established.

I therefore have a question:

Have we ever lost any mathematics?

Are there mathematical areas that have simple collapsed, having been accepted widely as true, even rigorous? I would like to rule out the case where an area has been rendered unimportant by the development of different techniques. In that case the results still hold, but are no longer as interesting.

Edit: 10/5/12

Some great discussion on math overflow, including one serious candidate, Italian algebraic geometry.

The 2×1 rectangle is not one of mathematics most celebrated shapes.

Yet it is so much more flexible than the more common square.

Even better you can cut it in half on the diagonal to make a 2×1 right triangle,

which has the beautiful property that it is a 5-reptile. Five copies of it come together to make a larger version. Repeating this gives the Conway Pinwheel tiling, which has triangles occurring in an infinite number of directions.

Yet the 2×1 rectangle is a lot more common in life, just go into your local hardware store:

Using the diagonal cut triangle and uncut rectangles, Vinay Gupta designed the hexayurt,

a small house that can be built from 12 sheets, without waste. In contrast to geodesic domes, that cannot be made from sheet materials without making many cuts or wasting material. Here is one:

and a plywood one:

Hexayurts have become one of the standard accommodations at Burning Man:

or look at this map, the red dots show the location of the hundreds of hexayurts at last years event.

Vinay set me the challenge of making larger domes using these shapes. The hexayurt itself suggests that hexagons will be important, and we can put two 2×1 rectangles together to make a square. Squares and hexagons come together to form the truncated octahedron.

This obviously would not work as a dome, so we must cut it. There are two natural cuts that can be made. One perpendicular to the 4-fold axis, and one perpendicular to the 3-fold:

So we have two new larger domes, the tri-dome and the quad-dome:

What is really cool is that both of these domes were made for Burning Man last year:

Tri-dome:

Quad-dome:

One neat thing about the truncated Octahedron is that it is a space-filler. You can use them to tile 3d space. We can therefore bring quad-domes together to make even larger structures, like this one:

The phyllotaxis spiral is one of the classical forms of mathematics, and there is a wonderland of resources available online both images and explanations. The basic idea is to put points round in a spiral with the same angle between each point. This gives a family of forms:

Note that, as the angle changes the dots sometimes pack in better than others, this can actually be studied and the best packing is related to the golden ratio. The points in this spiral are placed down in order, so we can associate each to a whole number:

Now when I see a lit of numbers like this, I want to pull out the primes, see what pattern they make:

There are some hints at patterns, lets expand out, and look just at dots:

There seem to be spiral arms which are richer in prime numbers than others. We can analyse things further by colouring each number depending on its prime factors. The more prime factors the lighter the number, giving the image for the start of this post:

Now there is a clear pattern, light and dark arms spiraling out. Can we understand this pattern?

Think about the construction of a phyllotaxis pattern we turn the same angle every time, that means within a particular pattern we can find other phyllotaxis patterns. The one at twice the speed, three times the speed and so on.  For example we could dive our pattern into two patterns each with twice the rotation angle. This gives:

All prime number (other than 2) are odd, so they must lie on the subspiral corresponding to the odd numbers. In addition it turns out that the spiral arms that we see are related to the Fibonacci numbers (themselves closely linked to the Golden Ratio). The particular curves we see relate to 144. Here is the spiral given by multiples of 144, pulling out just one such curve:

Note that in the prime factor picture this curve gives a very light line as every number in it is a multiple of 144, and 144 itself has 6 prime factors (three twice and two four times). Taking the multiples of 6 instead of 144 (which gives us several of these curves as 6 divides 144) we see another pattern of lines that are light in the image:

More importantly the curves next to these ones give numbers one more or less than a multiple of six. Every prime number has this form (all other numbers are multiples of 2 or 3 or both). This gives the curves of prime numbers we saw.

So by considering the construction of the initial image it begins to reveal its secrets. Yet, just as with the primes on their own there seems to be plenty of mystery left for investigation…

The book grew out of a single tweet from Vinay Gupta, it is a mix of dreams, plans, fears and wild hopes, yet all carrying a sense of reality. Although it much be said that such is the pace in which predictions stale that the future today already looks different in many ways to the futures painted in this book. I am proud to have two essays included in this book. You can buy the book, take a look at the essays online, see further developments of the project, or discuss on twitter, to help us all build a future we deserve.

I was thinking about this as I read the post on The Renaissance Mathematicus  talking about the birth of HistSci Hulk, sworn enemy of anyone who is wrong about the history of science (a noble and dangerous quest). This might seem to be the opposite position to the one that I gave above. I have felt the sting of his corrections myself, luckily in private not public! It is not opposite, in fact it is the essential counterpart. Being wrong is positive, but only as it helps on the way to better understanding. Reading about how the concepts of gravity were starting to come together before Galileo, and that he did not experiment by dropping things from the tower of Pisa, does take one further. Yet this does not make the original story worthless. It introduces the idea of gravity, the sense there was a change in understanding and  Galileo, himself.  The correction builds far more happily on this knowledge than it would standing on its own. For this to be effective, of course, we have to accept that stories (especially much loved ones) can be wrong, and more to the point we ourselves might be wrong.

I believe this is actually the great strength of the scientific method, and mathematical proof. Not that they can be used to show things are right, not even that they can show things to be wrong, but that they give a framework to persuade someone they are wrong. They help to develop understanding faster and further.

So do not get embarrassed when you are wrong. Do not get defensive. Learn to embrace it, be grateful, admit it. Then you are learning.

“It is better to open your mouth and learn that you were a fool, than to remain silent and never know.”

Some other takes on the same idea come from the inventor James Dyson and the author Kathryn Schulz.

1 (one) word
10 (ten) words a haiku, a sentence or a tweet

100 (hundred) words a paragraph, an abstract, a newsitem

1000 (thousand) words an article or blogpost

10,000 (ten thousand) words an essay or short story

100,000 (hundred thousand) words a book

1,000,000 (million) words an epic, Proust’s “A la recherche de temps perdu” is 1.5 million, the complete Harry Potter Saga is just over 1 million.

10,000,000 (ten million) words  an Encyclopedia, the 2002 Britannica is 44 million

100,000,000 (hundred million) words  a large Encyclopedia, like the Yongle Encyclopedia from fifteenth century China

1,000,000,000 (billion) words  Wikipedia (actually over twice that)

Then there is a gap…

10,000,000,000 (ten billion) words

100,000,000,000 (hundred billion) words

1,000,000,000,000 (trillion) words

10,000,000,000,000 (ten trillion) words

100,000,000,000,000 (hundred trillion) words gives you the internet in 2008

So perhaps soon the internet will surpass the work of a single man. The great french author Raymond Queneaux:

10,000,000,000,000,000 (ten thousand trillion, ten thousand million million, ten million billion) words  the word count (assuming 10 words per line) of the complete text of “Cent mille milliards de poèmes“

Having exploded outwards, it is not time to come back down, through encyclopedias, books and stories, back to tweets and the word:

1/10 (tenth) of a word a letter

1/100 (hundredth) of a word gives you a line segment which has an interesting property, it can itself be divided.

1/1000 (thousandth) of a word gives you a shorter line segment, allowing you to dive as deeply as you wish theoretically, in practice you will dive surprisingly quickly through atoms, protons, neutrons and quarks to the lower limits of our understanding.

Where it relates to how the bending strength of wood changes depending on the number of knots. From this lovely book, that I found at the local second hand book shop during Samuel Hansen’s recent visit to Fayetteville:

Which, is full of other equations and models, such as this one:

which is then explored for several values of n.

Some of the tables caught my eye just for beautiful way that they present information:

Finally, its not just equations, there is also a collection of patterns, along with the intriguing chapter on Structural Design of Sandwich Construction (probably not what I am thinking about):

All this points out to me, once again how mathematics can be a powerful tool to help study anything. I know that when it comes down to it this is really just the well established link between mathematics and engineering, but, as a material, wood is so much more accessible and visceral than, say, concrete. For some a book on wood might even answer the eternal question of “How am I going to use this?” but it does at least show that quintic polynomials really do come up in real situations!

but 2+2 = 1…Mod 3

Despite this surprise, we actually all use modular arithmetic regularly, quite literally on a daily basis. When we consider six hours after 8am, the answer is not 14, but 2pm. Well you could argue for using a 24 hour clock, but no one would claim that 3am on a Tuesday morning is really 27:00 on Monday (well apparently some do, thanks to kuromagi on reddit for ref) In these cases we are not counting as we usually do, but counting on a circle mod 12 or 24. It is not hard to see that we could do this with other numbers. if we do decide that 2+1 is 0, and not 3 we are now working mod 3. In this case 2+2 is 1, as is 2*2. We can put together a small table:

Showing what happens when the values for the column and row are added together. We can make the same table for multiplication:

I have to admit these table are a little boring, we can make things more interesting by replacing the numbers by colours. As we are working with modular arithmetic we know that the range of numbers we will come across, lies between 0 and the value we are using for modulus, so we can map these onto some circle of colours. So work mod 151 we get a new table for addition:

Using the same system of colours we can do the same thing for multiplication:

Which is starting to get interesting. We do not need to stop there, we can produce an image where the row number is taken to the power of the column:

This looks a little jumbled, in fact it seems to have very little structure at all. This is not very useful if our goal is to make pretty images, and on this blog that is normally the goal, but it other areas it turns out to be incredibly useful. The process of modular exponentiation is an essential part of public key cryptography, one of the technologies that allows secure communication over the internet. The jumble and lack of pattern that we can see is a sign that modular exponentiation is a good method to use to jumble things up. if there were structure that could be used to help decrypt the messages!

Returning to images, lets make a big version of the multiplication image, mod 1583 (you need to click it to get the full effect, scaling the image down blurs out a lot of structure):

Another option is to make an animation. what happens as we move the modulus value:

There is plenty to study in these images, for example, the curves that can be seen are approximately hyperbolae as they occur when is some fixed value. The central star point occurs in the middle of the image, and there are further stars at 1/3, 2/3, 1/4, 3/4 etc. Can you work out why?

The appearance of hyperbolae perhaps implies that other curves might be possible. What happens if we consider ? An obvious guess from this formula would be circles and we indeed get (for 151):

Playing around a little further this image comes from :

These images are certainly worth repeating for 1583 (again the details get blurred out, so click the images to see the full detail):

To finish let us consider something even simpler. Taking the value of a square to be this will always give a value between 0 and 1. We can then colour again, and animate with and going from 5 to 0:

I first came across these patterns in the December Issue of notices of the AMS, I have always been surprised how little they have been explored. This post is my attempt to do a little to correct that.

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.

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 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.

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.

This post is not trying to do anything clever. It is making a statement that seems self-evident:

There are three ways to gain understanding of the world:

• Personal experience
• Systems of rules
• Stories

All are equally important, and each has its strengths and weaknesses.

The important point is not the content of the statement but the stating of it. This is not just something that feels correct (to me) but something that feels fundamental. This mirrors one of the quests of mathematics to find the simplest statements on which to build the whole subject. I have my suspicions that the same thing would not work completely here, though writing the “Elements of the Academy” with this as one of the axioms might make a curious exercise!

This axiom maps onto the world of academia. The Sciences are primarily concerned with the use of rules to understand the world; the Arts centred on the creation of objects that attempt to transfer personal experience; and the Humanities write, dissect and try to understand the stories of the world.

All three areas, of course, do and should take advantage of the strengths of the other two methods as well as their primary concern.

The story

As a mathematician I obviously come from the grand tradition of finding rules to understand the world. For much of human history this was known to be rather limited in its scope. It was applicable to commerce, certainly; but also to questions of measurement, and to the study of the stars and music. Then, with the acceptance of arguments based on infinitesimals and the genius of Newton and Liebniz, the models of calculus opened up a vast array of phenomena to understanding through rules. It was so successful that many started to believe that it would eventually explain everything.

I do not believe this to be the case. Chaos theory shows that even perfect models can be severely limited by small, unavoidable, measurement errors. The work of Gödel and Turing shows that even in the purely theoretical world, there are unanswerable questions. Some even believe that as fundamental a system as arithmetic might contain contradictions. Before we even get to these hard limits we must deal with the soft limits imposed by the great ideas that we have yet to have.

Unfortunately, or fortunately depending on situation and personal preference,  the world offers many questions that we cannot answer with a systematic, rules based approach. Questions we cannot ignore. I wanted to define for myself the other options, and place them in some imagined framework.

The personal experience

I don’t believe I have said much here. It is, as I stated, self-evident. I also think it is important. It has been useful and practical to me. So, if you have managed to read this far, I thank you, but ask one further thing. Think about it yourself and see if it is a useful for you too.

Acknowledgements

This post grew out of a string of tweets, out of which grew very valuable discussion with  Colin Wright (@ColinTheMathmo) and Daniel Colquitt (@danielcolquitt), on twitter and elsewhere.

When I started working on the details for the tri-dome I realised I had made a bad assumption (thinking that the form was geometrically pure). This means that some of the details in my original write up were wrong. All a little embarrassing. Ironically, I might have missed a form that does actually work, had I not made the bad assumption. The shape, like the hexayurt, starts with a hexagonal based pyramid. In a traditional hexayurt this lies on top of a hexagon of vertical walls. Instead of this we attach a square to three of the edges and the classic hexayurt triangle (isocoles triangle with base and height the same length) to the other three. We can look at what happens as the pyramid is moved away from the ground, while the edges of the shapes rest on it:

This does not give a great building; there are holes. The holes are triangles and two of the sides have a fixed length. The final edge changes length, starting long, and ending short. We know we can fill the holes with classic hexayurt triangles. Two of the edges are the right length we just need the third. The length changes smoothly as we raise the roof, and starts shorter and ends longer than we want. Here we can apply the intermediate value theorem, so the correct length must be passed. As a mathematician I would stop there, the system works; however people are building the things…

So to calculate the correct angle for the square sides of the model we can look vertically down, calling the angle of the square face θ, (and assuming that the boards we are using are 8′ by 4′) needing as the classic maths problem asks to “find x”.In this case

,

we want so:

Solving the quadratic:

Which gives an angle of about 49°, and the height of the roof (assuming 4′x8′ panels) is , just over 6′ at the edge and 10′ in the centre. We can use these, and useful facts about general tetrahedra to calculate all the angles between faces by using the lengths of their edges. If you want to follow the details yourself, you need to add vectors to get some of the edge lengths, then use the Cayley-Menger determinant to find the volume of the tetrahedron, and then the generalised Sine rule to (halfway down this page) to give the angle.

Technical details for TriDome: angles to nearest half degree, lengths to nearest inch (assuming 4'x8' panels). On the left the angles between faces and point heights, on the right lengths and angles of the base.

Technical details for QuadDome: angles to nearest half degree, lengths to nearest inch (assuming 4'x8' panels). On the left the angles between faces and point heights, on the right lengths and angles of the base.

Finally here are the hexayurt models (rhino 3dm and vrml formats) of the hexayurt, H13, TriDome, QuadDome, plus a couple of others, including a very large one.

[A recent collaboration with Vinay Gupta, available as a pdf]

We are all products of our environment, so education is one of our best chances of producing a better human race in time to do something about our world’s plight. Our instinctive approaches to educating our children are rooted in our deep ancestry and our more recent cultural accumulations. As we see all around us, instinct and culture are failing us. Our inability to correctly model our world and act on our conclusions endangers us all.

Our ability to believe in our models rests firmly on our affinity for mathematics, yet centuries of breakthroughs in mathematical thought have not been broadly integrated into our culture. Although the fruits of pure mathematics – nuclear physics and digital computers and networking – more or less define the modern age our basic regard for the practice of mathematics has not increased in keeping with its importance, nor have our educational practices reflected the changing role of mathematics in the world. Cryptography is the backbone of all commercial use of the internet, and while hackers draw endless media attention, do you know the names Rivest, Shamir or Adleman?

Although mathematics is at least as old as agriculture our mathematical heritage is not as treasured as other cultural links with the distant past. Correcting our cultural bias against mathematics is an intergenerational struggle. In sport, art and music we encourage appreciation by non-practitioners, but interest in mathematics is expected to be confined to experts. Prejudices like if it’s not hard it’s not mathematics have interfered with our ability to appreciate or even identify mathematics.

Quilting and other forms of textile design, have some overt mathematics, counting and measuring, but making satisfying repetitive patterns uses the mathematics of symmetry. Tetris uses the tetrominos for pieces. Part of the satisfying regularity of the game is that the pieces aren’t arbitrary – all the possible shapes are there. Traditional card games lead to many areas of mathematics, but the deck itself is rather arbitrary – why four suits, rather than five? We need better artifacts to train thinking.

Games
Set In comparison to a standard deck, the Set card game is very ordered, having 81 cards (3x3x3x3). This forms a regular-yet-surprising deck, including every possible card for four choices of three options, and thus has the same sense of completeness as the Tetris blocks. Hands are matched all-same or all-different, and even very young children catch on quickly and can compete against adults!

Doodling You can make your own mathematical games on squared paper, or just play around with ideas. For inspiration you need look no further than Vi Hart’s videos.

Puzzles
Rubik’s Cube The ubiquitous Cube was the definitive puzzle of the 1980s. The 3x3x3 plastic puzzle encapsulates substantial group theory, and is solved by discovering or learning algorithms. Guides for learning how to solve the Cube have improved a lot over the years, it’s easier than ever to solve.

Penrose Tiles These two simple shapes fit together to produce an endless array of different patterns which never repeat and never run out. The puzzle pleases when decisions made earlier come back as you find you have to retrace your steps to continue laying the tiles. Beautiful patterns and shapes result.

Toys
Lego is the universal solvent for technical professionals. Everybody played with lego, and everybody describes how formative lego was in shaping their capacity to plan, execute and make. Modern lego has tended towards branding itself as a toy rather than a building system, but large boxes of basic bricks are still available. You can even bend it!

Zometool Want to see four dimensional space? This toy gets you about as close as is humanly possible, and you just have to build it. It is also brilliant for exploring three dimensions beyond the right angled system of Lego.

Polydron A simple idea, shapes that clip together at their edges forming a hinge. Mathematically they can look at how geometry jumps from two dimensions to three, what will you make out of them?

Meccano Another classic old toy that should not be underestimated. Metal and bolts vs. machined plastic. The long standing “Meccano people vs. Lego people” controversy can easily settled by buying both.

Scratch The easiest way for children to make software, taking their first steps into the source code that will run our lives. Scratch has excellent support for sound, graphics and even video, and is free.

Further Resources
Martin Gardner Ask mathematicians what got them into the subject as there is a very high chance that Martin Gardner will be mentioned. For years he talked puzzles, games and even broke new mathematical results in his Scientific American column. He left us with books stuffed full of curious intriguing and meaningful mathematics.

The Museum of Mathematics opens in 2012 in New York, this will be a mathematical wonderland, giving an intuitive glimpse even into many corners of mathematics. The website is packed with videos and resources.

Edmund Harriss & Vinay Gupta, Cloughjordan, 2011
with the kind support of Django’s Hostel

Type the phrase “scientists find the gene for” into Google and 68,000 results appear. Most of the hits are about human beings – which is a pretty impressive number, given that we have only 20,000 genes altogether.

Francis Galton: The man who drew up the ‘ugly map’ of Britain   Steve Jones

We have become used to numbers swirling about us, we talk so often about their power, that we forget that on their own they are meaningless. Meaning must be added, and we need to be careful when comparing, as he did, two numbers that come from different contexts.

We give meanings to numbers in many different ways, sometimes only using some of the abstract properties. House numbers make full use of the ordering on numbers, but No. 23 does not combine with No. 41 to make No. 64.  Yet think about how we teach number. Nearly every primary school has a number line, it might start with one apple, two bananas, three oranges. Yet, while one apple plus two bananas might be one smoothie, it is certainly not three oranges. So remember the old saying!

The ultimate form of abusing numbers is the bogeyman of numerology, diving into the abstract world of mathematics and jumping out again in different contexts to pull some conclusion out of thin air. So I might surprise you by coming to its defence.

A classic method is to turn a written idea or just a single word into a number. Then look up that number to see what other words come to the same value. Perhaps, for some, the meaningless of the number stage is preceisely the purpose. The ideas connected in this way will have no obvious connection. The game then becomes finding something that draws them together. Not to see something of cosmic significance, but to stretch the imagination and get creativity and thoughts flowing. Creating a space for creative randomness.

So to bring this to some form of a conclusion, remember to be careful with numbers; but do not be afraid of them. Think about, play with and subvert the meaning that they are given. Perhaps you might even get lucky in your random connections and realise something about yourself or the world.

When I entered the room for the third time I was a PhD student in London. I had the practical purpose of finding a book I needed, but I became entranced. It became a regular place to visit, gaining familiarity and comfort. During hard times in my PhD and later jobs in London it acted as a refuge. At some point earlier memories returned.

Old-School Foyles, but not the mathematics room. Do you have an image?

Of course, the first time I visited I had no idea that this room would become part of my personal mythology. I do not know how old I was,  I cannot even remember the context (a family trip to London?); but I do remember the room, standing out even from the magical L-space¹ that Foyles used to epitomise. Years later I returned. I was an undergraduate at Warwick and my love affair with mathematics books was truly beginning. Not just for the knowledge they contain, but for a beauty that I feel but not find words for. Part of this beauty is the esoteric language of their titles, the language that puts so many off mathematics but, it must be admitted, entices others in:

• The Arithmetic of Elliptic Curves
• Sphere Packings, Lattices and Groups
• Buildings and Classical Groups
• Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras
• The Geometry of Schemes
• Analytic and Algebraic Topology of Locally Euclidean Parameterization of Infinitely Differentiable Riemannian Manifold.
• …

Some books I simply gazed at, others I bought simply for the magic of their titles. Those titles echo in my memory. Today some have become trusted friends, some sit mysteriously on my bookshelves having resisted numerous attempts on their secrets, others turn up like old acquaintances when I visit book shops.

I did that recently, I was once again in Foyles. The mathematics section had moved once more. It was once again in a familiar room at the front of the building on the third floor. Had they come home (albeit sharing the space)? But wasn’t the old maths room on the second floor? To my shame I could not remember clearly. I had grown used to the room being gone; it was a shock to find the details of my memory so weak. My internet skills failing me, I could not even find a record that gave the floor, worse, I could not find mention of the room at all. So I decided to write this.For me it makes concrete memories that seemed routine at the time, but now hold great importance, but perhaps I am not the only one? Maybe there are others who have fond memories of this room. If you do find this and remember please share your memories of this odd, impractical but special room.

I mourn the room, but  do realise that some things have to change. Today the nature of the book itself is changing, and with it the bookshop. Just opposite Foyles the space that used to be Borders bookshop is now taken up with TK Maxx. With the ability of the internet to deliver information and,  electronic readers finally usable, the paper book finds competition it has never had before.  Yet the bookshop, as I adore it, has been under threat for a long time. Borders itself along with Barnes and Noble represented the first assault, opening up the bookshop and making it easy to navigate. Then Amazon opened things up further, making it possible to easily find any book in print. Yet great bookshops, like Foyles, have survived, I have faith, there will be changes, but some of what we love in these stores will survive and perhaps some of what will be lost needs to go. Is it such a bad thing that cheap dectective and romance novels will no longer force trees to be cut for their paper?

For the moment therefore I  try to regularly  visit the bookshops I love and buy books from them. Not just for the books themselves but as a support for those wonderful shops.  It makes a good excuse anyway!

Footnotes

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As a regular visitor to Foyles I learnt certain routes around the building, mixing the stairs and elevators. It felt that a tiny deviation from the correct route could leave you in a different place entirely. It was occasionally a shock to realise that two points, that I had thought were in completely different parts of the building, were actually just around the corner from each other. Terry Pratchett describes this best with the concept of L-space, that all libraries, and bookshops in the world are connected both in space and time and, with the correct path, you can navigate to any of them. In the Discworld version of the burning fire of Alexandria a hairy arm is seem amongst the flames rescuing some of the greatest works.

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. There were plenty of examples to be found (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:

A simple, classic puzzle is to give two shapes and ask if there is a way to cut one up so the pieces can be rearranged into the other. This game might seem to become silly if both shapes are the same;  if we insist that the new arrangement must be different the game becomes interesting again. Think about it, can you come up with ways to cut up a square so that the pieces can be formed into two different squares? Here is an example, not with a square, but with a rhombus:Having the same shape has an advantage. Think about the letter p below, it is part of the blue trapezium, when we rearrange the tiles the p moves with the shape. As the two shapes are the same we can think of this new p within the original rhomb. We can now repeat the process as many times as we want. In this case, it might be a little unsatisfying, however, as the next step for our p would cut it into two different pieces, as it lies on the edge. So where is it safe to put a p so that it will never get cut up? To answer this we have to follow the cutting lines, and a beautiful pattern emerges:The p would be safe within any of the pentagons, but if it crosses any of the edes it will, eventually be cut apart.

Puzzle: Can you work out the difference between the green and the blue pentagons? (Hint: it relates to the dotted and solid lines in the earlier pictures).

Studying what happens when we can move points or objects around in a space (in this case moving p around a rhomb) is studied in a part of mathematics called Dynamical systems the particular example here is called a Piecewise Isometry  (see this paper for a more formal account of their history and study). I have studied these systems myself, and recently submitted a paper looking at the behaviour and number theory that occurs within the pentagon generating system shown above (take a look! It has lots of pictures as well as more formal mathematics).

As you might have guessed from my preoccupations part of my interest in these systems is the pretty images that they produce; this system is particularly rich. This leads to the image at the top. You can take any rhombus and cut it up in a similar way. Take any rhomb (as shown below) and rotate until the side of the rhomb lines up with the top. This will leave a triangle and a trapezium that can be moved back on top of the original rhomb:Additionally this gives a system where the rotation on the two parts is the same, just around different points. You have to be a little careful, but you can use this to give a system for any angles. For any of these systems we can ask the question: Where is it safe to write p? Every angle gives a different pattern, and tiny changes in the angle leads to large changes in the pattern, however the patterns do relate to one another in some ways, as you can see in this video: