Just three letters are left out: j, k and z. Of course z does turn up in the number system in Zero, but curiously all three of these turn up a lot in the list of fake large numbers in English. Think of “kajillion” or “zillion”.

It is interesting how long we have to wait for several common letters, especially c. We can visualise this by plotting the smallest number for each letter. As the numbers get so large we use a logarithmic scale so the smaller numbers don’t become invisible.

In fact the logarithmic scale makes a lot of sense, after all we have more letters available every time we have used one. We can look at how many letters are needed for all the numbers up to 31:

A further reason for a logarithmic scale is that, as the numbers get larger the number of words that are needed also declines. We start at a quick pace, all numbers up to twenty have a new word. Yet after twenty we just have twenty-one, twenty-two and so on. No new words until thirty, then forty and so on in tens up to hundred. From there we do not need a new word until a thousand and then a million. We can plot the largest number expressible given the number of distinct words:

With standard practice this will continue in a straight line (on the exponential plot) until we run out of words. They do go a long way though.  There is a small question that arrises here though. That is the size of a billion. This has been used for both 1,000,000,000 (a thousand million) and 1,000,000,000,000 (a million million). In each system a trillion has a corresponding meaning. In the first 1,000,000,000,000 (a million million) in the second 1,000,000,000,000,000,000 (a million million million). The same for quadrillion, quintillion and so on. The first system is sometimes called the short count and the second the long count. In the image above I used the first, more common usage. Yet the second version has a distinct advantage if we associate a million with 1, billion with 2, trillion with 3 and so on we can then consider what happens when we multiply. A million times a million gives a billion, just as 1+1 = 2, a billion times a trillion will give a quintillion, found easily by 2+3=5. This ability to add to work out a multiplication relates directly to the effects of using logarithms. It is not easy to come up with an simple way to find the answer when multiplying large numbers in the short count.

So, suppose we use the standard number rules and allow 35 words (that is all the regular numbers and the -illions up to heptillion), what is the largest number we can count up to? In the short count it is nine hundred and ninety nine heptillion, nine hundred and ninety nine sextillion, nine hundred and ninety nine quintillion, nine hundred and ninety nine quadrillion, nine hundred and ninety nine trillion, nine hundred and ninety nine billion, nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine or:

Which is pretty large, while also show why we might prefer a numeral system over words for these big numbers. The long count does even better with nine hundred and ninety nine thousand nine hundred and ninety nine heptillion, nine hundred and ninety nine thousand nine hundred and ninety nine sextillion, nine hundred and ninety nine thousand nine hundred and ninety nine quintillion, nine hundred and ninety nine thousand nine hundred and ninety nine quadrillion, nine hundred and ninety nine thousand nine hundred and ninety nine trillion, nine hundred and ninety nine thousand nine hundred and ninety nine billion, nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine or:

Which is a lot bigger. We can however make an even more powerful system without really bending the rules that much. We already have smaller numbers multiplying larger ones, for example a hundred thousand. What happens if we allow each power of ten number after a million to be multiplied by all smaller ones, without repetition. With this system we can consider a hundred thousand million before we need a billion in the long count. Note that to make this fit I am arbitarily denying the use of ten. In this system therefore a trillion is not a million billion, but a billion billion, a quadrillion is a trillion trillion and so on. With the same 35 words we can now get to:

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hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine quintillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine trillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine quadrillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine 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ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine trillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine quadrillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine trillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety 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or:

This massive growth comes as we are now using the exponential of an exponential. Why not try to come up with even more efficient methods that need even fewer words? The binary system is worth investigating, especially if you consider why it might be efficient and how this links to its use in computers. You could also venture into the delightful game of constructing larger and still more massive numbers.

We are used to reading mathematics, we are also used to hearing it spoken in lectures. I can think of few examples of the natural way to combine these. Why do we never read mathematics out loud? There are some good reasons for this, much of the symbology of mathematics was developed as visual and so has only a bad translation into speech. More importantly mathematical reading is very rarely linear, we step back from a theorem to a definition only partly remembered, jump forward to look at the corollaries before diving into the proof.

Yet speaking words has a power that simply observing them in your head cannot. I tried for years to enjoy Paradise Lost yet got nothing from it, until I heard a comment from Phillip Pullman that it needed to be read out loud, and its beauty opened up to me. Why can’t mathematics benefit from this? We have our share of great writers, Donald Coxeter, Tim Gowers, Donald Knuth, Douglas Hofstadter, Rudy Rucker, indeed John Conway‘s papers are often created by writing down his spoken words. I am of course obliged to mention Lewis Carroll and Martin Gardner.

Many people, myself included, are interested in how art can be used as path into mathematics. The visual arts and music are well represented, even mathematical symbols have been considered for their aesthetic qualities. Writing feels relatively neglected, yet is intrinsic to the actual practice of mathematics. So I have a question:

What mathematics would you choose to read out loud?

I am especially interested in passages that work when read, even though they are written for a mathematical audience. The more esoteric the better.

If we can find some great ones, then perhaps we could even persuade someone with performance skills, who is also interested in mathematics, to read them out. Yes Vi Hart I am looking at you!

To design my wedding rings I started with digital and algorithmic systems. Not for any particular reason, but I am good at them and enjoy the sorts of control they both give and take away from me. The computer makes such methods easier and faster helping develop helping take the ideas out of my head. Here is my design:

The problem then is how to get the ideas and forms out of the computer. There are several options. For anything two dimensional we now all have incredibly accurate printers at home. Even in 3d pretty great options are starting to emerge. Yet these technologies did not feel right for wedding rings, perhaps in part because they felt too easy. Just to make the task harder I also wanted something that retained some sense of the design process. Not just a way to make this particular ring but something that could, in principal just as easily make any of the other rings that I did not choose. In other words I wanted a process, something that could take in a weave pattern and give out a ring (or in this case two). This was the point when it is good to have friends and this is the process worked out with (to be honest mostly by) Eugene Sargent.

Firstly the design shifted a little bit to allow for a casting process and also give a stronger ring:

The next trick was to actually make the woven pattern. We did this using copper wire:

The problem was we kept on getting lost as we tried to follow the individual strands of wire through the weave. The solution was to consider the crossings not the strands. The strands are labelled 1 to 8, and when they cross they also swap numbers. So if 1 and 2 cross the strand that was 1 becomes 2 and vice versa. This might sound complicated, but it means that you can forget the individual strands and only need to consider their current position.

We were reinventing the wheel in this as it is an ancient technique for creating braids. For example it is used in the classic hair braiding technique where the hair is divided into three strands and first the right and then the left strand are brought into the middle. With the strands labelled 1, 2 and 3 this would correspond to swapping 1 and 2 and then 2 and 3.  The idea of considering crossings is also used in the mathematical study of braids, called appropriately braid theory, and can be used to produce images of braids simply by describing the crossings.

Using this method we could describe a wide variety of braid patterns and reliably weave them. This would produces a woven copper ring:

That can be wrapped round a blank:

We then used epoxy to fill in the overlaps. The finished blank was then used for sandcasting. Making the mold:

That could have molten silver pored into it:

Revealing a fairly rough version of the ring:

With polishing and filing the cast rings ended up like this:

The key is that this process could be used for any weaving pattern without significant change. In this sense it is parametric, with the parameter being either a pattern or, more usefully, the abstract listing of the crossings that describes the pattern. On the other hand the process that uses the parameter is simple handwork combined with the ancient technology of sand-casting. The computer, though a useful tool in the design phase, is not a necessary part of the process.

I have always been interested in the possibilities for patterning of weaving, which can be described as the entwining of one dimensional structures in three dimensions in order to produce two dimensional objects. To start as simply as possible with just two threads there is only really one thing to do:

Three threads give a more interesting (and classic) braid:

With four threads the number of options starts to expand rapidly, and drawing braids out by hand is a rather time consuming task. I therefore wanted to create a simple system that generated weaving patterns from some easily described information. The object I started with is known as a permutation. This is an operation that takes an ordered list of objects and puts them into a new order. We can consider the two braids above in these terms, the first takes two threads, and simply swaps them. The full pattern comes from repeating this. The second pattern is slightly harder to read, but it becomes clearer if we consider a single unit:

Here you can see the top goes to the middle, the middle to the bottom and the bottom goes up to the top. We can encode this permutation as 0-1-2, with 0 corresponding to the top and so on. The new position of each element is given by the element to the right. 0 (top) goes to 1 (middle), which goes to 2 (bottom). The final term simply returns to the first, so 2 goes back to 0. Note that 1-2-0 will therefore give exactly the same structure. On the other hand, 0-2-1 will give the structure turned upside down.

With a simple way of describing a collection of braids it was a lot easier to start exploring the space. Even making things by hand could be done in a more systematic way; however with a little coding many patterns could be quickly explored. Starting with a 3d image of the braid above:

With four threads the family of permutations becomes a lot richer. With three threads, choosing where the 0 thread will go determines the whole structure. So we can choose to start with 0-1, this leaves only 2 to give 0-1-2. Or we can start with 0-2 and be forced to use 0-2-1. With four threads the 0 thread can go to three different things. Each of those leaves two options. The final choice is still fixed. There are therefore six different cycles, as with the three thread example however, reversing the order of the cycle simply flips the braid left to right.

0-1-2-3 (flipped 0-3-2-1)

0-1-3-2 (flipped 0-2-3-1)

0-2-1-3 (flipped 0-3-1-2)

With five threads the number of choices increases again. The 0 thread can now go to four different positions, there are then three choices then two before the final position is determined as before. This gives 4*3*2 = 24, as before we need only consider half of these as reversing the order simply flips the pattern. This leaves these 12:

At this stage we might realise that this method has some limitations. It is a useful tool to explore patterns (0-2-4-1-3 and 0-3-2-1-4 show particular promise). Unfortunately more symmetry is coming in, for example 0-1-2-4-3 and 0-1-4-3-2 are the same patterns flipped top to bottom rather than left to right. It is harder to spot this from the permutations on their own, a top to bottom flip swaps the roles of 0 and 1 with 4 and 3. Other patterns are related to each other by a 180 degree rotation. A more insidious problem comes when we consider the weave rather than just the pattern. Consider 0-1-2-3-4, 0-1-2-4-3 and 0-1-3-2-4, each of these has the same woven pattern stretched in different ways.

We have also only considered permutations called cycles where each thread travels through all the positions. There is something satisfying about these but we might be missing out. For example with four threads we could have 0 going to 1, 1 going back to 0 with 2 and 3 swapping in the same way. This, would not be too interesting as it would just give two copies of the braid with 2 threads. Instead we could consider 0-2 1-3 (note no dash between the 2 and the 1) which gives:

In fact we can write any permutation (reordering) as a list of cycles in this way. Another way to consider permutations is just to consider 2-cycles where two elements flip. For example the 3-cycle 0-1-2 can be considered as 0-1 followed by 0-2. In this case 0 is flipped to 1 which is not effected by the second flip. 1 itself is first flipped to 0, the from 0 to 2 by the second flip. Finally 2 is not effected by the first flip and then goes to 0. Overall this gives 0-1-2. Studying the structure of permutations like this both provides examples for group theory, but also essential tools. The idea can also be used to avoid some of the problems we identified above in the study of weaving patterns. The mathematical study is called braid theory.

Despite these flaws I was able to use this system to explore and find a pattern I liked:

Here is my final design, taking in to account some thought of the methods available.

Find out how it was made real.

Two Englishmen stand in the highlands of Fiji, gazing up at a hill high above the village they are staying in. As there is not a lot else to do, they decide that it makes a great goal and head off. Of course they do not know much about the land beyond what they can see, so they head off in a straight line. Dropping into valleys and struggling up to peaks they make some progress, but after several hours the peak still lies in the distance, and they return home. The next day, fed up with goals they simply head out for a walk. Ambling along the watershed ridge, as they have had enough of steep hills, but like the view they find themselves at the top of a distant peak. Looking back they realise they are standing at the top of the peak they has set out for the day before.

This story is true, but to me it has also become a personal myth. Like all good myths it gives a space to take in ideas, give them a good shake to see what falls out:

• The myth of modernism: The straight path to the mountain is always the best.
• The myth of post-modernism: All paths to the mountain are the same.

My intention in calling both things myths, is not to dismiss them, but to start to think of them as fundamental ideas. To me the story of the mountain trumps both these positions. Yet what I really want to start with another myth entirely:

• We all have mythologies.

This myth can be easily shown to be true, simply by a clever definition of mythology, so instead let me discuss my own. It comes from many places, some of the most significant influences include:

• Mathematics
• Christianity
• Terry Pratchett
• Douglas Adams
• Collapsanomics
• Icelandic Sagas
• The Malvern Hills

These things give a great deal of framework to my life, colouring my reactions in ways that I cannot predict. When presented with something new they will effect how I behave for good or bad. They help me work on when I need to use a direct approach, and how to meander when it might not work. To use a computer analogy they are the fundamentals of my code. To keep that story going, I therefore think that it is important to think them through and debug them, hopefully before those bugs come out in a crisis. They are, in many ways, my religion.

It has become quite common for people to see the immense harm and trouble that religion has caused throughout the world and see the solution as being no religion. Yet I do not think we can get away that easily. Even in mathematics we have to take the set of axioms we use on faith, we cannot show that they cannot admit a contradiction. We have to be careful where our ideas come from. My friend Vinay Gupta recently said on twitter:

The problem that we have in the west is that we thought the cure for Bad Religion was No Religion and it’s left a generation lost.

@leashless

I agree. The solution to the problems of both bad and no religion that I have tried to develop is to identify and analyse my religion, trying to take responsibility for it and live by it.

Acknowledgments

This post started life in a discussion with Bembo Davies (from whom I stole the point about taking ideas into myth and metaphor and shaking them), Michal Woźniak and others on the role of totems in the future at the Edgeryders conference. At the time I talked about mythological maps to our emotions. As you can see my thinking has developed from being told maps to the need to develop our own. Many of the ideas have been made clearer in my own head by conversations with Vinay Gupta, as well as a study of his twitter stream.

1) Teaching is impossible

The idea of teaching implies that you can be the active party in someone else’s learning. This is not really the case if you want to go beyond a little rote recitation and rule following.

2) Impose yourself

Once you have accepted that you are engaged in a fool’s errand get arrogant, unless you are confident that you can persuade, cajole and trick people into learning for themselves, you will not be able to. In order to do this you must be able to gain some control, getting a classroom or individuals to listen to you. Without some form of control you will be ignored or even humiliated. Once you can gain control, however, please do not stop there. Many do, and they become the legends people complain about for years to come. Instead…

3) Do less

Remember that what you do really does not matter. It is what your students do that matters. If you have opened up a class discussion and it is going well and on topic, let it be. The best state for anyone learning is when they go for it on their own, the teacher silent.

4) Confuse and take risks

Now we are into the essential, but dangerous skill. There is certainly bad confusion, but there are good forms too. Again this is about what the student does, more than about you. The simplest thing is to simple “be less helpful” but you can take it further and take a risk. Make your students confused, make them fail, it can really help their journey to learning independent of you.

5) Learn

I have used the term students throughout this piece, but that is wrong, try to drop it from your thinking. Take every chance to learn from the people you work with, make it a two-way engagement. Also never forget to consciously hone your craft. You might have explained how to do a certain problem hundreds of times, is there a new way to try?

One of the biggest barriers in mathematics is that one often has to unpick a previous understanding in order to go further. This is true both of individual learning and the progress of the subject as a whole. I feel that these issues, combined with the desire to assess and make learning visible have caused some deep issues in the way we teach mathematics. The canonical work on this topic is, of course, Lockhart’s lament. In many ways I hoped that the disruption that technology is bringing to established models of education might change this. I believe that effective mathematical education is of great importance to the future will we live in.

Today, I felt cold fear through my veins as a long building realisation crystallised. There is a serious danger that it is already making things worse. The depression really kicked in as I found myself changing my opinion of Sal Khan dramatically. My old opinion, was that he was of great value in making information of reasonable, and often good, quality available. Though perhaps this could only ever replace part of the role of a teacher. An interaction between Khan and Wired blogger/physics educator Rhett Allen, made me change that. Let me set the scene. A couple of math teachers made a video talking about some of the issues in one of Khan’s videos. To Khan’s credit this lead immediately to changes to the treatment of that topic. Dan Meyer and Justin Reich responded by suggesting a competition to find other issues in Khan’s videos. The spirit was to help, by providing free peer review, improve the quality of material within what has become a standard resource, rather than to criticise Khan’s work.

Rhett Allain’s response was quite simple, and did not seem to even deal with the subtlties of pedagogy. Rather it looked at the use of vectors, and pointed out some, perhaps slightly subtle, factual errors in Khan’s treatment. Perhaps because of Rhett’s profile in Wired this video received a personal rebuttal. In many ways what Khan says there is correct. The way he presents the idea of vector certainly makes things easier for the particular problem he is working on. Yet, I also feel he has completely missed the point of Rhett’s criticism. This is one of the situations where making somethings easier can perhaps introduce issues that will actually make things harder later. On the other hand Rhett and I might be wrong and, in any case Khan has every right to defend himself. In particular I do not want this to become a personal attack. He has personally done amazing work. My fear comes instead from the authority that he has gained from this. You can see it in the comments that people made in response to the “Correction Correction”.

Khan’s magnanimity in responding to what was clearly aimed at discrediting a valuable online tool is admirable.

This highly reeks of the criticisms targeted towards Wikipedia from supporters of the traditional encyclopedia. Well, look what happened to Britannica – their physical editions have been discontinued. Those who cannot embrace change are destined to become obsolete.

FATALITY! Rhett Allain, go to amazon.com and buy a rope and a stool! Sal, you are the best!

These are, of course, comments on the internet and should not be taken too seriously, and there are other defending Allen. I believe they do demonstrate something, however. A more worrying example (to me) came from a previous discussion the Khan Academy and education on the edgryders site. There Alberto Cottica, who is a serious and subtle intellect, with experience around education and communication defended Khan saying:

Dr. Khan’s videos are pretty much the best shot at access to an outstanding math teacher I have ever had

So my crushing fear on this is that people might not be able to recognise the best mathematics teaching. Thus enabling an even smaller number of “master teachers” to dominate, perhaps including Khan. Even worse it might be that the system cannot even choose the best candidates for the role of “master teacher”. My hope instead is that we can move to a new model, summed up by another edgeryder, James Wallbank in what I would love to make a mantra:

Everyone needs to be taught to learn, and to learn to teach.

“What have you been doing?”
“I was at a the “Living on the edge” conference?”
“Where was it?”
“The Council of Europe”
“What was it about?”
“Ummm?????”

The edgeryders conference was clearly about something, but actually pinning that down in words gets tricky.  In part this came from the range of people there, so perhaps that is a good place to start. In that range to me there seemed to be three archetypes^*, if not warring then at least engaging in collaborative competition, sometimes in a single individual.

The first, fired up with ideals and energy, certain that the world could be changed if only the right policy could be found and implemented. With plenty of ideas on what that policy should be (after all certain things worked incredibly well for them).

The second, brilliant and cynical, hiding a little measure of social discomfort behind trolling and alcohol. Knowing that there were powerful ideas and possibilities that could be unlocked, but just as certain that many of them would be broken by bugs many times. Their wisdom gained from long hours testing and fixing code.

Finally a slightly less common individual, a little harder to pin down. Grumpy certainly, but also emotionally grounded, skeptical of the new, often because they had seen it before. Interested far more in the human challenges than the technical.

Of course no one was exactly any of these three, I would guess that I was mostly the second with some aspects of the third. Yet towards the end of the unconference something interesting started to happen. The three archetypes began to combine, the energy of the first, driving the brilliance of the second through the subtlety and human touch of the third. Many sessions got to powerful fundamentals. This made something else clear, the human and emotional are absolutely key to the ideas being discussed, while also the hardest to deal with. The problem, if anything becomes worse in a technical audience. These problems are messy, defying language. What can be said can sound wooly or just obvious, yet trying to focus and define language can just empty it. We need ways into these topics, is was therefore slightly sad, that the third archetype had the poorest representation of the three groups.

In this spirit therefore, perhaps a technical explanation of the event is not the right approach. Here instead is a poem:

Heaven in Hell’s despite
Peter Blegvad

This rocket’s going nowhere
It’s travelling so fast.
In one end goes the future,
out the other comes the past.
And we are on a mission as we hurtle through the night,
to build a Heaven in Hell’s despite.

This rocket’s going nowhere,
the hull is full of holes.
No one navigating
No one at the controls.
No one said it would be easy
on our maiden flight,
to build a Heaven in Hell’s despite.

This rocket’s held together
with string and chewing-gum.
We don’t know where we’re headed,
don’t know where we’re coming from.
But we’ll know it when we see it,
when we see the light.
We’ll build a Heaven in Hell’s despite.

* BACK TO POST  I am aware that I am reinventing the wheel a little here, these three archetypes seem to have something fundamental about them. My goal is in part to record some initial reactions. The role of the third archetype in the community is perhaps to establish this sort of mythology that will help drive things forward. I hope to return to and develop that idea. The Dark Mountain project and Bembo Davies’  concept of Human Rites, both follow similar ideas (and are probably well ahead of where I will get to!).

We have many problems in education, and too often the response is to look to at technology and say “Oooh Shiny!”. The excitement over Khan academy being a great example. Yet Khan academy to me has little to do with education. It is fantastic at putting information into people’s hands, but cannot give them the skill to recognise what they need to know. In this it resembles the internet itself.

Think about the internet. Sitting here in bed, without moving much, I can go out to a massive chunk of humanity’s knowledge. It is a profoundly different fact that today a kid in Africa lucky enough to have a computer can access more knowledge than a professor traveling round the world’s best research libraries could when I was that age. This should have a huge impact on education. In the face of this sea of information every fact you learn in school, if it is not wrong, is useless.

Yet good education and good teachers have never focused on facts. In addition teaching is not a technical problem that can be solved by clever young thinkers. The gap between theory and practice is very large. Teaching is about performing magic in someone else’s head. Changing the way that they think. Sometimes that requires a positive approach, encouraging the person learning. In other cases it involves telling someone that they should do better, violently rejecting good work as something great is possible. A great teacher will know the time to sit back and let someone bash their head against a problem, getting lost and frustrated. Then find the exact time to ask the question that reveals the crack that allows them to open up the whole thing for themselves. As Dan Meyer says “Be less helpful“.

Yet a great teacher will also get these things wrong, probably most of the time. Remember you are trying to change someone’s mind, that is never a simple act. As a teacher you might not even know if you got it right or wrong. As a student you might not realise how right your teacher got it until years later, if at all. This does not mean that we should not try to measure and observe learning, but does mean we have to be careful. Great theoretical models and shiny ideas might work well only in what we can measure or just in theory.

To take education forward, let us look at the old models, from school and university to apprenticeships and martial arts training. Let us consider how Sushi chefs have traditionally trained, how farmers pass on their knowledge of the land to the child who must care for it one day and how clowns are introduced to their world. Let us approach these old models humbly and see not how we can disrupt them, but how we can enable them. Learning their lessons, wisdom and failures as we bring to bear our clever new ideas.

• 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: