We begin with a list of curvesKeeping the idea of a list we can create circles traveling along another curve you are free to define, varying radius as you go.

The next one does not look so interesting perhaps, as it is just a frame from an animation, click to see the whole thing. It is a version of the dots from the previous image traveling along a track, but now using differentiation to make sure they stay touching. A second animation, uses the direction of the curve at each point.

Some more calculators doing interesting things, for example distorting a grid of circles:Or showing a vector fieldor showing the first family of cubic equations to have their directions analysed by calculus, from Newton’s tract of 1666:

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For curvahedra, however, with their curvature, you can have two different paths of shortest length going between two points. Think about going round either side of a hill.

This gives a two sided polygon, or bigon, and a cute collection of Curvahedra. The bigon balls:

This effect of curvature can be seen in the night sky. Objects with significant gravity, such as galaxies and black holes significantly curve the space around them. Light, wanting to travel the shortest distance goes passes round the object in many different ways, rather than passing straight through, in a process called called gravitational lensing.

This can be seen as a repeat of the same star several times, for example in a astronomical feature called Einstein’s cross:It can reveal black holes and magnify distant galaxies, and provided an important piece of evidence for general relativity.

You can get your own Curvahedra pieces to explore!.

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So what other spheres or near spheres can be made? A good answer from those with a little geometry are the regular polyhedra. These are the 3d shapes where every face, vertex and edge looks exactly the same as every other.These shapes provide the climax of Euclid’s Elements, and the proof that they are the only ones is up there with Hamlet, the ceiling of the Sistine Chapel or the Taj Mahal as one of the greatest human achievements, but you can make this one yourself from paper:There is a reasonable argument that each of these must be close to a ball due to the high symmetry. This is a little stretched for the tetrahedron, but we will give it a pass. Can we do anything else with these pieces? To make a sphere we want to have the same amount of curvature everywhere. This does not require that every shape created have the same number of sides and the same angles at the corners, but that seems to be a useful place to start. Then at least every face has the same total curvature. In addition as all the angles on an individual curvahedra piece are the same, the angles around a corner should be the same. These conditions lead us to the Catalan solids, or Archimedean duals, the shapes where the model looks exactly the same from every face. Thus every face in a Catalan solid is the same. Here is one made with Curvahedra:

but that is not a sphere, what went wrong?

Looking at an individual pieces we see that the problem is, the curvature is not spread equally over the shape, one end is more bent.

The same polyhedron with flat sides shows where this might come from. The edges of the shape are not all the same length (but all curvahedra edges are).That does leave two Catalan polyhedra though, the rhombic dodecahedron and rhombic triacontahedron (with 12 and 30 sides). For these every edge is the same length.

They use 3 and 4 connectors (dodecahedron) and 3 and 5 connectors (triacontahedron) arranged to make rhombs

This gives us two new balls!

The triacontahedron is particularly pleasing, being a large ball,, as you can see when it is compared to the classic ball:

So by reasoning we were able to discover something new, but have we found everything? Maybe other close to perfect spheres are possible, what can you find?

Find your own Curvahedra pieces here.

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Taking a close look, each piece has five arms, and they are equally spaced around so the angle between two arms must be 360/5 or 72 degrees.

The interior angles of this triangle are all the same so we have 3*72 = 216. Yet from geometry we know that the the interior angles of a triangle always add up to 180. What has gone wrong?

Here is a 60 degree triangle (note the pieces have six arms, so the angle between neighbours is 360/6), can you see the difference?

Unlike the first triangle this lies flat on the table whereas the first curves away. The difference is clearer if we complete all the pieces around a corner for each.

Going further the five triangles come together to form a ball, while the six triangles would keep on spreading, we won’t be able to complete that sheet.

What we are seeing here is the curvature of the surface we are making. The triangle with 72 degree angles can be said to have an excess of 36 degrees. The greater the excess the more it curves. Look at this triangle with 90 degree angles (for a total of 270 degrees, an excess of 90 degrees), the curvature is very clear:

Completing this creates a smaller ball.

This new ball has eight faces, each with a 90 degree excess. Adding all these together gives a total excess of 720. The first model has 20 faces, with a 36 degree excess, and again a total of 720. Lets think about the model with just three triangles around a corner:

The total angle is 120*3 = 360, so the excess is 180 degrees. If the pattern holds we should need 4 of these triangles to make a ball, and indeed we do:

In fact if you take anything that is like a sphere, take the angle excess on every face you will always get 720. For a more complex example take this model:

This has eight triangles and eighteen squares, and all the angles are 90 degrees. For the square this is normal the total interior angle of a quadrilateral should be 360 and 4*90 is 360. So there is no angle excess. This leaves the eight 90 degree triangles once again giving 720. Also notice in the model the square faces are flatter with the curvature occurring at the triangles. This gives the model the shape of a cube with rounded edges, rather than a sphere.

This result is called Descartes’ Theorem and it is a special case of the Gauss-Bonnet theorem, both are closely related to the Euler Characteristic. These theorems stand at the heart of topology and differential geometry.

A natural follow up to this is to ask what happens with a shape with two little angle (an angle defect). For example the sum of the angles of a quadrilateral should be 360. What happens if we take a square (4-equal sides and angles) with 72 degree angles. The sum is now 72*4 = 288, which is less than 360. This creates a saddle:

The saddle is said to have negative curvature, and connecting up more and more squares, like this does not create a ball connecting up on itself. Instead it gives this wavy surface that grows faster and faster, modelling a hyperbolic plane, all these images are the same object!

Final note: The curvature discussed here is actually called Gaussian Curvature, and is a property of the surface itself not the way it fits in space. For example consider this cone:This is covered with equilateral triangles with 60 degree angles. So although it looks curved the geometry is flat, the triangles all have no angle excess. In other words if you investigated distances just on the surfaces of the model they would be the same locally as those on the locally flat plane of triangles given above. You can only detect the change if you loop back on yourself round the cone. The only exception is the tip of the cone. Here you can see a piece is left hanging.

The same thing happens when you bend a piece of paper, you change how that sheet lies in three dimensions, but not what happens on the sheet. You can even use this to work out how to best hold pizza. On the other hand the Gaussian curvature, discussed above, does change what happens on the surface. The angles of triangles can be measured without leaving the surface. In fact this might have been part of Gauss‘ motivation. He wanted to work out if the earth was a perfect sphere, but did not have access to space. In other words he had to take measurements just on the surface of the earth.

These ideas had even greater importance with the work of Einstein. General relativity assumes that the three dimensional space (or the four dimensional spacetime) that we live in is itself curved. In fact that curvature is related to gravity and explains how gravity acts at a distance. This huge idea fundamentally changed our understanding of the universe yet we can start to appreciate it with a simple toy, which you can get for yourself here. Another way to explore the geometry is with crochet from Daima Tamina’s beautiful book.

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My latest project has been several years brewing. A system of hook together paper pieces that can make all sorts of interesting geometry. I have used it to engage people with mathematics, to teach mathematics and am now making it available to everyone on Kickstarter.

Help me spread the word of this fun math project! It has already been featured in an article by Alex Bellos in the Guardian, and written up by MTBoS regulars Sam Shah, Christopher Danielson and Joe Schwartz.

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Linear algebra is one of my favourite areas of mathematics. Its a simplification but you could say that the things that mathematics does well are small numbers and straight lines. The rest is just clever ideas to covert other things into those. As the mathematics of straight lines and flat surfaces, the importance of linear algebra should be clear. Its techniques are also very fast when done on a computer, allowing live motion in video games. This lead to every computer having a GPU, essentially a special chip for linear algebra.

Within linear algebra a central object is the linear transformation (that can be encoded as matrices) and the subspaces it preserves given by the eigenvectors. This gives some powerful tools to break linear transformations into pieces that can be studied more easily. As well as eigenvectors, however, linear transformations that do not have negative real eigenvalues preserve other families of curves. Curves that are taken to themselves by the transformation. These animations show how these curves change as the matrix changes. Giving a glimpse into the detail of what linear transformations do. These are of particular interest in dynamical systems where these images so some of behaviours possible close to a fixed point.

This animation shows a pair of eigenvalues changing from complex (creating a rotation and scaling) into real values. Can you tell where it happen? It uses the matrix

with s running from 1.05 to 4.05.

This animation shows a hyperbolic fixed point (attracting in one direction and repelling in another) changes into a attracting equilibrium point, using the matrix

with s running from 1.05 to 4.05.

This animation shows a transformation changing as both the eigenvalues and the direction of the eigenvectors, using the matrix

with s running from 0.05 to 3.05.

This animation shows a transformation changing as both the eigenvalues and the direction of the eigenvectors, using the matrix

with s running from 1.05 to 4.05.

This animation shows how a shear changes with different eigenvalues, using the matrix

with s running from 0.05 to 3.05.

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Then in May Alex had the great idea of getting into the growing trend for adult colouring books. The intricacies of designs allowed, was perfect for a book of mathematical images. It seemed that this idea caught on and we had interest from publishers, but it came with a big caveat “Can you make it by June the 15th?”. We slightly hesitantly accepted and got to work. We made a book in a month! Of course if you have seen stuff on this blog you will know that I have been secretly preparing for this book for years. So secretly I was not even aware of it myself!

The result is Snowflake Seashell Star, (Patterns of the Universe in the US). A collection of images to colour and rules to follow to relax with and hopefully to enjoy.

This book is particularly satisfying as it fits one of my most dearly held agendas. To help the deepest and most beautiful ideas of mathematics be more accessible to everyone. Too often we hide more advanced mathematics for fear of causing confusion. Yet, in an analogy I stole from Marcus du Sautoy, we do not do the same in English where material, like Shakespeare, that can be studied for a whole lifetime can be introduced in primary school. How do we help access the equivalent literature of mathematics?

There will always be many answers but hopefully this book will provide one. People begin with the patterns, working on them in detail as they colour. There is plenty of potential to make discoveries about the underlying structure but no pressure. Any discovery that is made is a significant achievement. Having engaged with the pure forms, a curious colourer can then read in the back of the book to discover a little about what it was they were studying. I hope that many will have made their own opinions before finding out how to see what they were doing in delightful new ways. With careful use of google they might be able to dig further, finding more about the mathematics behind the images they love. Or perhaps not, again this is pressure and testing free! Everyone can choose their own way to engage and enjoy the material.

For those for whom the notes at the back are not sufficient and the google searches are frustrating I hope to cover some of the material on this blog. If there is a topic you would particularly like to hear more about let me know in the comments below!

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Start by clearing your mind.

Now imagine one dot pop into view.

A second dot joins it. Let the two dots flow around each other, rotating and getting closer and further apart.

Now a third dot, creating a line or a triangle…

Keep on adding, with each addition try to see all the dots, find a shape you like…

This is about having fun and playing with math, which often sounds a little:

This is not the holy grail, it is not even a challenge to bring into the classroom. Teachers have too many challenges, sometimes the challenge is just to get through the day without messing up too badly. It is an encouragement to relax and have fun, yet remember that this fun is part of your teaching prep!

Playing with maths can often start with going back, returning to something you know well, and trying something new, testing an idea. If it fails try something a little different, or go back to work out how it went wrong. If it works, can you try everything? Mathematicians can say everything and really mean it! Even then do not settle, go back with your new knowledge and try something new. You might notice once you have started you cannot escape! You can always just stop. This is play not work. Though it might not be relaxing, just as playing a sport is exciting, fun and cathartic but you put effort in.

This is why this can build into your teaching, once you have fun you have a chance to help your students have fun. If they have fun they will put far more effort in than if you have to push them. Also I do not feel that mathematics has a huge number of facts, but isolated they are not that useful, going back and playing with ideas helps build the dense web of connections that really drives understanding.

General strategies are great, but it can be hard to know where to start, I will describe two tools:

- Analogy and the concept of same/different (mathematics is the world’s greatest metaphor!)
- Breaking rules! (yes mathematics is often about creating them, but also about changing them and seeing what happens).

To get further, we need an example, and not one that will lose half the audience just with its title so…

Three dots, are they the same or different? They are in different positions, but are the same shape. We have to be clear what we mean.

Now we take pairs of dots, we can spin them around and pull them apart. We could say they were the same if they can be moved on top of each other. Yet to define that precisely we have to use most of plane geometry. We have not even counted past two and we already need that!

Getting to three the line and the triangle, different in ways that the pair of dots can never be.

Lets change tack, we have been looking at how the same number of dots can be different, what about how different numbers of dots can be the same?

These patterns for four, six and eight have some similar features. How might we describe those precisely so we can identify other ones? Saying that the numbers are all even is an obvious way to do it, but maybe they also share something with this:

Like the earlier examples nine dots drawn like this form a rectangle (specifically a square). Following this definition we can define prime numbers (technically composite numbers!).

Here are another collection of dot patterns that share features, one dimension, two dimension and three dimension, and at this point reality gives up on us. Yet we really went past our page after two, we can use the notions of analogy to push further. We know the next pattern will have sixteen dots. For example we can make this image, with lines to show the structure. Can you find the eight cubes?

With a little work from here we can work out that an nn-dimensional cube has 2n (n-1)-dimensional faces. So we know very little about 172 dimensional space, but we do know that a hypercube in that space has 344 faces! Playing with some of these tricks we can get this:

There is a lot more to discover in this image. If you are interested in getting a version send me an email, I am looking into options.

Lets move to the other trick, breaking the rules. Mathematics is made of rules, yet there is not one rule that is not broken somewhere else in mathematics. For example this might make you uncomfortable:

**7 + 7 = 2**

**2 + 1 = 2**

If I say instead that seven months after July (the seventh month) is February then the first makes perfect sense. In this case 7+7 is still 14 but 14 is the same as 2, we have modular arithmetic.

That trick will not work for 2 + 1 = 2. Yet in Chemistry two hydrogen molecules combine with an oxygen molecule to create two water molecules. There is an even greater rule, though one that has been enshrined in legend. Yet this image shows what happens when we divide by zero (at the centre)!

(the mathematical trick is to use what is called the Riemann sphere).

In conclusion playing with math can happen with the simplest structures and lead to a variety of thoughts and adventures. No one should be shy of having a go!

Here is a neat animation from my play:

I have a list of some other materials to inspire your mathematical play, and there is a whole world of examples in Sue van Hattan’s book Playing with mathematics. That should be available for pre-order soon!

Many other have explored the idea of simple pictorial versions of numbers, often using prime factorisation. With dots and circles, with monsters, or even to make a game. Although my personal favorite are these dots, with their illusion of simplicity.

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Focus on that final one. A farmer with five sheep in one field and seven in another can combine those two numbers to know he has 12 sheep without getting them all in one field to count. This is the heart of mathematics. We can use the equals sign and other tools to generate new facts from old without having to go back to the evidence. We know that putting sheep together in a field and then counting will give the same answer as counting and then adding.

This is actually a very profound idea and a formal version of it is at the heart of category theory^{1}, even considered by many mathematicians, to be abstract nonsense. Yet it is something we do every day. When we use numbers we intuitively take into account the properties we can use. Go back to that sum:

**5+7**

When both fields counted sheep we could combine them. If one counts sheep and one cows then we have a units problem, though we could combine to get 12 animals. Even with the same units we can not always use all the properties. Offices 5 and 7 do not combine to give office 12, May and July do not combine to give December. It is in a real sense five months added to July. Though if we add five months to December we can back to May, which opens up several questions!

I could go into the formal details of these different systems, that is not the point. Instead we should realise that even dealing with something as mathematically nonthreatening as the number 12 there is a huge amount of complexity in how it is used.

This issue is at the real world problem problem. To show how a particular technique is used we must cut away all the translation issues, we must even cut away all the other mathematics until, however important and sincere to begin with the problem feels trivial and fake.

At this point you might hold up your hands in despair. To me this gets to the heart of the teaching of mathematics. There are (at least) two distinct skills:

- The mechanical system of converting one thing to another using an array of symbols
- The translation system mapping those moves onto the world.

We can emphasise the mechanical (the approach of most school systems) or we can emphasise the translation (the approach of Conrad Wolfram and others who argue that computers should do the calculation). Personally I sit on the balance. Without the translation your models become just intellectual games. Yet without the models you have nothing to translate to^{2}. The challenge of teaching mathematics is to balance between the two. It is hard especially under all pressures that “things would be simple if…”. Its hard but on occasion mixing the two right can set off a virtuous cycle where advancing in one motivates effort in the other.

**1** BACK TO POST My article Category Theory for Designers has a little bit more on this topic.

**2** BACK TO POST Just translating to a computer might have potential, especially Keith Devlin’s approach of using computer games to develop intuition. Yet if not careful it can miss the understanding of what the model actually does so becoming a magical incantation.

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and you get the Lissajous curves:

These are a generalisation of the circle, that appears when a and b both have the same value, or simply for the function

.

Now consider another parametric function:

.

Set and , this means that

which is:

So the parametric function satisfies the definition of the circle , which surprised me as I did not think that the circle had a parameterisation by such rational functions (one polynomial divided by another). In fact the lack of such a function was the source of questions and proofs on math.stackexachange.

So why do we go through all the trouble of creating sin and cos (ok they do have other uses!). One reason comes when we look in more detail at each parameterisation. When you just look at the points given by the function you ignore how fast those points change position as t changes. If we space points out equally by t we get the following images:

With (on the left) we get the points equally spaced around the curve, but needs to use the whole real line with changes in t moving the point less and less as t increases. You can analyse this in more detail by looking at the differential of the parametric function. In fact that is how I sized the circles in the image.

There are also some interesting connections between the parameterisation and the function arctan (think differential).

The fact that none of this makes the proofs on math.stackexachange incorrect is left as an exercise!

I can’t leave without showing the Lissajous curves with their dots. You can also explore these patterns for your own parametric functions.

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That’s the corporate design spin on the new logo for the Twitter Math Camp, but for an audience of mathematics geeks a little more detail might be appreciated. So here goes.

The image of the logo uses a technique to visualise functions from the plane to the plane. We are familiar with visualising functions from one dimension to one dimension. This can be done with a line plot, the y axis value of a point giving the value of the function at that point, for example :

You should go and play with this in desmos. Functions from two dimensions to one dimension, for example can be shown using three dimensions. In this case we look over a range of possible values of x and y and create a surface where the height is given by the value of the function.

Here we hit a snag. Our eyes and brains have never been trained to see beyond the three dimensions our reality provides. This is really quite unfortunate, as a lot of mathematics only really starts to get going in three or four dimensions. Luckily space is not the only dimensional thing we can perceive. There is also colour. We can redraw the previous image colouring by height, from yellow at the bottom to red at the top:

Now the information is in the colour we can remove the height, just showing a flat image with the function determining the colour at a point:

We can now take advantage of the dimensions of colour for example a second function can leave the colour the same but make it lighter. So we can use a new function that takes in two numbers and gives two numbers back the first number gives the colour as above, the second lets the colour fade with the function creating a fade as we go further from the bottom left corner:

So we have solved our problem, by using two spacial dimensions and two colour dimensions we can visualise functions that take in two number and give two numbers out. In fact we can go further as our perception of colour is a three dimensional system, with independent amounts of red, green and blue. These actually do not model colour at all, as in the world each wavelength of light is independent so colour is infinite dimensional. Our eyes, however, only have three sorts of receptors so the infinite dimensional space is projected onto a 3 dimensional one. It is quite possible to have higher dimensional perception of colour with more receptors. Most birds and some fish and turtles have been found to have four channels of colour perception, leading to an experience of colour far richer than we can imagine (but still far short of reality).

Just dealing with two dimensions though does turn out to be incredibly powerful, thanks to complex numbers, which make many mathematical problems easier and are used throughout science and engineering. They can be expressed as two dimensional numbers over the real numbers we have been using so far, through the Argand plane. So a function that takes a complex number to another complex number is 2d to 2d (over the real numbers) and is precisely the situation we have described. The process of visualise complex functions in this way is called domain colouring, and I have linked to a google image search, though be prepared for intense psychdelia (in the name of math). To explore the topic in more detail the book Visual Complex Functions by Elias Wegert is beautiful and worth getting just for the images.

To get to the TMC logo, we have to leave behind using the dimensions of colour, however, and use a different method to colour our image. We are dealing with a function that takes in two (real) numbers and gives two (real) numbers back. We can use the two numbers we get out to find a point on the plane, a point in an image, and use the colour of that image at that point. The result is called the *pullback *of the image through the function, as we can consider that the image has been literally pulled back through the function.

For the logo I started with a pattern made up of the letters TMC: then made the letters fade out as they went away from the centre:with the image in place I could start to explore complex functions through this notion of pullback. Generating huge worlds of potential logos:

For these images, however, the letters TMC are hard to pull out. A better candidate is :The final image used this, though breaking the purity a little the three middle copies of the letters were removed to be replaced with an undistorted version. Of course you can also switch the image, I hope Megan does not mind me using her copy of the iconic image from TMC13 pulled back through : As a final note you might notice that the final logo has a different colour to all the examples shown here. Just as colour can be generated from numbers we can also pull numbers out of a colour, change them and then go back to colour, in this case shifting red to blue.

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As you play…wait you did click and play didn’t you? **Go back and do so at once!**

As you play there are two interesting creative directions to try, exploring and mapping. In the first you just try things to see what will happen, in the second you try to gain control. For the first anything goes, for the second you often way to try the simplest thing you can. Talking of which ready to play again?

This time you were playing with a function from the plane to the plane. We can’t draw a standard graph for those without seeing four dimensions. I have tried and tried but can’t do that. So we have to be more creative, there are a few ways, but this works by showing how an understandable image changes.

You might have noticed that this does not always work, for some functions you enter the image goes wrong. For example try

and

can you work out why, and how this might be avoided (or used to advantage)?

To finish lets introduce one more chart. This one plays with parametric equations where both the x and the y positions of a point are described by functions. If you investigated the first chart you might have seen them already. Parametric equations often have a point moving at different speeds along the line, which you can see here, the function thinks all the circles are equally spaced:

Another example to try here is:

These notes, explore further working through some classic functions and starting to control the image, to me the balance between wild play/exploration and control/understanding creates the space where art can happen.

If you find functions you really like with the second method send me the details and I will create a laser cut version (demand permitting).

I have been working on the exploration of functions found here for some time, the ideas were originally developed along with David Celento and Brian Lockyear for a workshop at Acadia 2012. Sam Shah also used the ideas for student projects.

The main problem was that they were only implemented in Mathematica. Telling people “Here’s this cool idea, but you need expensive software to run it” just did not seem to work for some reason. Coming across the beautiful Desmos finally got me to create a more accessible version. Though there is no reason that I could not have made on in Geogebra and I plan to. Let me know if there are other formats that might be useful. I am also looking for help from teachers to connect these ideas to the classroom.

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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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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.

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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!

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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.

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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)

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.

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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.

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.

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.

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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?

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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:

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.

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