Word powers of ten

February 2, 2012

How do we understand the number of words on the internet? Its hard to even grasp how many there are, and the number is growing so rapidly. Trying to understand a similar problem, the size of the universe (or just the observable universe) Charles and Ray Eames came up with the classic Powers of Ten video. Lets try the same for words:

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.

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Posted by gelada

Polynomials in Wood

December 4, 2011

What has 1-x/2-6x^2+11x^3-7x^4+3/2x^5 got to do with wood? Like you until a few days ago I would have said “Probably nothing” then I came across this chart:

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:

N = \frac{PQ}{P sin^n \theta + Q cos^n \theta}

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!

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Posted by gelada

2+2 = 1? Patterns in Modular arithmetic

November 20, 2011

When someone is talking about the absolute truth of mathematics and declares that once you have defined 2 and +, then 2+2 must equal 4, there is a slightly glib response:

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:

+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

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

x 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1

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 x*y 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 x^2 + y^2? An obvious guess from this formula would be circles and we indeed get (for 151):

Playing around a little further this image comes from x^2 - y^2 +3 x y:

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 \frac{x \mod y}{y} this will always give a value between 0 and 1. We can then colour again, and animate with \frac{x \mod Q y}{y} and Q 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.

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Posted by gelada

The University Project: Stories and Science

September 25, 2011

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

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

The world needs story tellers and stories

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

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

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

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

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

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

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

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

The cultivation of knowledge.

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

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

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

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

1 Comment | General | Permalink
Posted by gelada

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