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.

1 Comment | General, Main Article | Tagged: , , , , , , | Permalink
Posted by gelada

Numbers are meaningless

June 28, 2011

Although not the nicest of men Francis Galton was also a bit of a hero of numbers, drawing them into the human domain, and developing ideas such as correlation. Unfortunately, an heir to Galton, Steve Jones did not employ the same subtlty in a recent article:

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

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

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

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

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

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

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

2 Comments | Mathematics | Tagged: , , , | Permalink
Posted by gelada

Mathematical Scales

November 6, 2010

Thanks to the move to the US, my son has a new piano teacher. He is playing at an advanced level, beyond grade 8 (for the UK audience), with pieces by Bach, Mozart and Chopin often ringing out. Yet for the last couple of months he has been taken right back to the basics. Looking again at simple techniques on how fingers hit the keys and going over scales.

I am in love with this idea of training, taking someone who has proved incredibly able in an area and taking them back to the most basic ideas. I started to wonder what the equivalent might be for mathematics. What exercises should we be giving to starting PhD students?What exercises could we ourselves try in order to gain intuition and insight into the basic workings of our subject. I have a first proposal, but am sure there are others? What do you think? Of the idea itself, or of suggestions of possible exercises?

Multiplication Exercise

Multiply all possible pairs of numbers from 1 to 99, that is 4950 different calculations. At a conservative estimate of 120 per hour (most will be a lot quicker than 30s, some will be longer!) that is just over 40 hours work. That could spread quite nicely over a month, maybe two along with other activities. It would be 40 hours of meditation on the most fundamental of mathematical operations, what might come from that?
Other suggestions

A couple of excellent suggestions from commentors in a lively debate on reddit:

1) Teaching, which of course is already a significant part of graduate training in the US, unfortunately less so in the UK (those being the two systems I have worked in).

2) Deep study of proofs, with mention of this beautiful paper of Dykstra.

16 Comments | Mathematics | Tagged: , , , , , , | Permalink
Posted by gelada

Unscheduled Post: The Silver Ratio

May 20, 2009

John Cook on The Endeavour has just mentioned the wonderful silver ratio. As this is probably my favourite number I can’t resist the chance to put up some pictures. The silver ratio: \Psi = 1+\sqrt{2}, is as John mentions the value of the continued fraction with just 2′s, it is also the larger solution of the equation \Psi^2 - 2 \Psi - 1 = 0. This goes directly into its geometric interpretations, as the diameter of an octagon and the size of a rectangle that gives a smaller version of itself when you remove two squares:

Oct_and_rect

Geometric interpretations of the Silver Ratio.

In terms of tiling the golden ratio of course has the Penrose tiling, with its five fold rotational symmetry, the silver ratio plays the same role for the Ammann-Beenker tiling, with 8-fold rotational symmetry:

Ammann_Scaling_bb

Version of my Ammann-scaling artwork

In fact if you find things with 8-fold rotation (Islamic art for example) the silver ratio will be lurking around. I have a personal theory that the silver ratio was as much in Christopher Wren‘s work as the golden. I have not studied it in depth, the floor under the great dome of St. Paul’s has an giant octagon. Anyone know any good studies that might mention this, it would be good to have evidence!

Silver_circles

7 Comments | Art, Mathematics, Unscheduled Post | Tagged: , , , , , , , , , , , , | Permalink
Posted by gelada

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