Thursday, June 08, 2006

Cube, Part the First

Today: something cubish. Bear with me on this one.

1D: If you have a little line, you can use it to represent any 'bipolar' situation, like the potential outcomes of tossing a coin. You could label the ends H and T, or, if you like a spot of binary, 0 and 1.

2D: If you have a square, the four corners could represent the possible outcomes of tossing two coins (HH, HT, TH, TT). You might label them 00, 01, 10, and 11.

3D: A cube has eight corners (000,001,010,011,100,101,110,111). Looked at in a certain way, it's a map of the eight possible states of a three-bit computer. You could use this computer to store the result of three coin tosses, or perhaps a letter from a to h. At any one time, your computer would inhabit one corner of the cube. The cube is its little world of potential: if you give it a different letter to store, or a different set of coin toss results, it moves to a different corner of its world.

4D: A 'hypercube', has 16 corners. (If you're curious about cubes in four dimensions, there's a lovely explanation here - scroll down to 'Analogies to Lower Dimensions' and enjoy.) A computer living here would have four bits - a semibyte.

The laptop I'm typing into has a modest 80GB of storage, which is about 687 billion bits. It lives in a 687 billion-dimensional cube, flitting from corner to corner like a fly in a box. Every stroke of the key sends it to another corner. Even when I stop, it flits through dimensions I'm not aware of. I can hear it. Flitting.

One can't help feeling pity for the poor thing...

And it was between such bouts of pity that I took to wondering what it might be like in 687 billion dimensions (invariably a good move if at any time you find yourself fed up of any form of compassionate state).

A spot of Pythagoras (or a quick sketch with a ruler) will tell you that the diagonal of a 1cm square is about 1.4cm long. The equivalent distance for a 1cm cube - in a straight line between opposite corners - just over 1.7cm.

If you're ever trapped inside a 1cm crouton, or a sugar cube, you'll always be able to stretch out to 1.7cm long if you need to.

Picture now a sugar cube in 687 billion dimensions, still just 1cm across. The distance between opposite corners? A little over 5 miles.

This exercise has helped me come to terms with the plight of my flitting friend; and maybe it can help you too. No more fly in a box visions for me. I see my sweet laptop soar on snow-white wings through miles of sparkling space, to liquid crystal heights and diaphanous digital depths, and I am at peace.

Olivia said...

You are actually insane. This is the most actually insane blog that I have ever actually read.

clare said...

hear hear! now lets have no more of that 'i'm the only sane person in the universe ' nonsense, you are quite mad. and you know it.

suf said...

insane perhaps, but uniquely interesting.

Anonymous said...

... and the volume of said cube....?

Just ONE cm^(687000000000).
Claustrophobic HELLLLLL!
Demented laughter follows.....

Jonathan

Bob Hu said...

I've read three of your blog posts so far and I love them. Your writing style is very clear and the conclusions you reach are amazing.
I actually wanted to tell you that if you used some of the things you've written on your blog as things to talk about on dates you would be irresistable to any woman worth your time.
A few months ago I showed what a hypercube was to my girlfriend and told her about a 64,000 core supercomputer that was connnected up as a 16-dimension hypercube and she loved it.
Give it a try and let me know how it works out.

Anonymous said...

I'm his girlfriend. It worked for me.