'curled' Dimensions?

[joekgamer]

I had heard a while back that extra dimensions were 'curled up' inside of others. The way I see it, the 4th, 5th, etc dimensions should be perpendicular to all of the others and effectively 'heading towards' istelf. Could someone clarify?

[CraigD]

I had heard a while back that extra dimensions were 'curled up' inside of others.

This isn't the usual rough description of the compact dimensions written of mostly in string theory.

 

The usual description, such as the one in the preceding wikipedia link, describes them as curled up (or "compactified") inside themselves.

 

The way I see it, the 4th, 5th, etc dimensions should be perpendicular to all of the others and effectively 'heading towards' istelf.

This is the way string theorists imagine and formally describe them, too.

 

Could someone clarify?

Physicist Howard Georgi put it memorably, I think, in this poem (which I first encountered in Michio Kaku's wonderful 1994 popular science book Hyperspace):

"Steve Weinberg, returning from Texas

brings dimensions galore to perplex us

But the extra one all

are rolled up in a ball

so tiny it never affects us"

More prosaically, I find this analogy in 2 dimensions helpful:

Imagine we have a one-dimensional space - a line-world. It describes most phenomena well, until in an experiment, we detect that somehow, stationary bodies have a kind of momentum and kinetic energy.

 

So we theorize that out 1-D line-world is actually a 2-D world, with the 2nd dimension too small to directly measure. In 3-dimensions, we can imagine that our line world is actually a thin cylinder, like a wire, and that our mysterious momenta and kinetic energies are due to particles circling it.

 

A key concept here, and one like the 11 and 26-dimensional ones common is string theory, is that a space need not have the same geometry. In our line/cylinder world above, its observable dimension is flat, while its compact one is spherical - traveling along it brings you back to the same point, as an the surface of a sphere.

 

A final caution: sting theory, while very popular among theoretical physicists, is mathematically dauntingly hard, and not yet demonstrated to most people's satisfaction to be physically true. I found Lee Smolin's 2006 The Trouble With Physics had a good discourse of what is attractive, and unattractive, about string theory.