Monday, 16 June 2008

Entry 21: in which We kick around a hypersphere

"Pater" says the boy.


"This pink string disc thing ... hey I like it, it's got a ring! Pink string disc thing! Or we could ping pink string, or ... "

"Yes, yes, what about pink string?"

"Well you know how there's this 2D chap that we plop onto a 3D world and then first he's outside his pink string disc thing and then suddenly he's inside it and it all seems very magical ..."

"Yesss ..."

"Well you said to extend that by another dimension and Bob would be my uncle, who is Bob by the way?"

"Never mind Bob!"

"Yes, well anyway I've given it a lot of thought and can you explain it to me please?"

Good grief, what is the point of planting seeds of wisdom when they wither on the vine if I may mix my metaphors a bit. I sigh but continue:

"Right, so essentially we have our magical ball of pink string, the one that won't run out, and this time we plonk Mr 3D on a small 4D hypersphere. Well just as Mr 2D on the north pole of a 3D planet looked around him and thought he was on a familiar 2D plane, so Mr 3D looks around him and concludes he's on a familiar 3D sphere.

"His job is just to walk around and around the (initially small) sphere, playing out the pink string as he goes. As he does so, the sphere gets covered with pink string and starts to get bigger and bigger, just like one of those 'biggest ball of string in the world' things.

"He keeps going and the sphere gets bigger and bigger, huge in fact, so that the horizon is a very distant thing. But he just keeps going.

"After a very long time he gets a very big surprise; one that is possibly in 'nasty shock' category. He was on the outside of a very large pink sphere, but suddely he finds himself on the inside of a pink hollow, and what's more, a hollow that gets smaller and smaller the longer he continues.

"Just as Mr 2D could not understand how he went from being on the outside of a disc to being on the inside of the same disc (which to us, from a 3D persepective, was perfectly obvious), so Mr 3D simply cannot understand how he went from being on the outside of a sphere to its inside".

A long pause.

"Which to us, from a 4D perspective is perfectly obvious".

Another long pause.

Finally he pipes up "I'd give a lot to see one of those 'biggest ball of string in the world' things."

Wouldn't we all? Mercifully I am almost certain I hear Mrs G's dulcet tones requesting my presence and exit stage left.


Millennium Housewife said...

Just a quickie, have you ever seen the Simpsons episode where Homer goes 3D? Really worth a watch in terms of what you're describing here. Wisdom comes in all guises!! MH

Ernest de Cugnac said...

Actually, MH, I did see that episode. It was good, and it looked they had fun making it.

Jolly Good Yarn Girl said...

All those Sundays in pews whilst my berobed son struggled through the pain of trying to stand still and look alive in a choir stall and all I really needed was to sit in my office and read this now and again...feels like enlightenment! JGYG

Ernest de Cugnac said...

Hi JollyGood, nice of you to drop in.

"Feels like enlightenment", but probably isn't! Thanks anyway.

Janelle said...

how timely of mrs g..she rocks. x janelle

Millennium Housewife said...

Hi Yarney! Knew you'd like it in here..! MH

Gayé Terzioglu said...

I tried and tried but couldn't get my head around (ha!) the pink string thing. How can you go round and round and round, you'd be always on the outside to continue going around making it into a ball and how do you find yourself in it at the end?
*perplexed* I am just NOT smart enough!

Ernest de Cugnac said...

Gayé - you are indeed smart enough. I insist. Have you read this post? It's the two dimensional case. Someone, on a plane, going round and round with the string, would always be on the outside of it, unless they were accidentally in a 3D world and on a sphere (which, let's face it, would still feel like a plane, and which we could perhaps describe as a hyper-plane). Thinking in 2D he cannot see how he could have come to be inside the string, but we watching can see how it could, indeed must, happen.

It's just extending that thinking to the 4th spatial dimension. Bear in mind that we all think in 3D, and in 3D it just CAN'T happen. But 4D creatures watching are falling about laughing at our bewilderment, because they can see, as we could above, that it must happen. With a hyper-sphere.

Another analogy might be the depiction of a cube on a 2D surface, i.e. a piece of paper. We, as 3D creatures can see that the drawing "means" a cube, but a 2D creature never could because they couldn't "project" it back into 3D the way we can. So if a 4D creature drew us a hypercube, or a hypersphere, we couldn't decode it properly because we lack that extra dimension.

All this raises real problems for a deity who might exist in 4 spatial dimensions (or more for all I know). How would they interact with us? How could we possibly understand those interactions?

That's why I like this quote from the great zoologist, Haldane: "My own suspicion is that the universe is not only stranger than we suppose, but stranger than we can suppose."

Gayé Terzioglu said...

Oh Hypercube!!!!!... Ok, now I got it. I think. I saw the movie The Cube and just recently The Cube 2: Hypercube.
I think of dimensions like I think of god. I have to see to believe. Is that so bad? When people shout at me that I'd go to hell for not believing, I say: If your god exists, and is all knowing as you say, then I have nothing to worry about.

Ernest de Cugnac said...

G - where have I been? I've never heard of the cube, but I googled it and it looks excellent. I shall have to track down the dvd.