Dimensional Venturing, Part 6 – Tiny Dimensions

“The Universe is much larger than is generally supposed.”  

What a great opening line, eh?  Decades later I still recall reading that in a technical paper about then-recent adjustments in the way astronomical distances were measured.

The authors didn’t know the half of it.  They were thinking in only three dimensions.  That’s so last-century.

If you read science articles in the popular press you’ve probably run into statements like this one from Brian Green’s article “Hanging by a String” in the January 2015 Smithsonian:

String theory’s equations require that the universe has extra dimensions beyond the three of everyday experience – left/right, back/forth and up/down…. [T]heorists realized that there might be two kinds of spatial dimensions: those that are large and extended, which we directly experience, and others that are tiny and tightly wound, too small for even our most refined equipment to reveal.

Tightly wound dimensions?  What’s that about?  And what’s it got to do with strings?

The “large extended” dimensions are the kind we discussed in Part 1 of this series.  The essential point is that (in principle) once you or a light ray start moving in a particular direction you can keep going in that direction forever.

Seems obvious, how else could it be?

tiny dimension 1Well, suppose that we bend one of those three familiar “large” dimensions around in a circle, as in the drawing to the right. Our little guy could walk straight out of the page “forever” in the X direction. He could walk straight up the page “forever” in the Z direction. However, if he tries to walk along the Y track perpendicular to both of those two, in a while he’ll wind up right back where he started.

That’s an example of a “tightly wound” dimension.

Because it makes the math easier, physicists usually don’t calculate the absolute distance traveled around the circle.  Instead they write equations that depend on the angle from zero as the starting point. Notice that 360 degrees is exactly the same as zero — that’ll be important in a later post here.  Anyhow, there’s reason to believe that the effective circumference of a “tightly wound” dimension is really, really small.

OK, having a closed-off dimension is a little strange but it’s just not real-world, is it?

tiny dimension 2Actually, our real world is like that but moreso. Look at this drawing where we’ve got a pair of perpendicular wound-up dimensions. The little guy on the Y track can go from Denver down to Mazatlan in Mexico and proceed all the way around the world back up to Denver. On the X track he’s going from Denver westward to Chico CA and could continue across the Pacific and onward until he gets back to Denver The only way he can travel in one direction “forever” is to go along the Z track, straight upward, and that’s why NASA builds rocket ships.

Back to the strings. Depending on which variety of string theory you choose, the strings wriggle in a space of three Z-style “extended” dimensions, plus time, plus half-a-dozen or more wound-up or “compactified” (look it up) dimensions.  If string-theory strings can wriggle in all those directions, then how much room does each one have to move around in?  We’ve all learned the formulas for area of a rectangle and volume of a cube — [length times height] and [length times height times depth].  To extend the notion of “volume” to more dimensions you just keep multiplying.

Back to the size of the Universe. You may think that just with straight-line space it’s pretty good-sized.  With those stringy dimensions in play, for every single cube-shaped region you pick in straight-line space you need to multiply that volume by [half-a-dozen or more dimensions] times [many possible angles] to account for all the “space” in all the enhanced regions you could choose from when you include those wound-up dimensions. The total multi-dimensional volume is very, very huge.

The universe is indeed much larger than is generally supposed.

Next week — buttered cats.

~~ Rich Olcott

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