Monday, May 4, 2009

It's orthogonal, dammit!

Alternate Title:  It's a point, dammit!

Also from the realm of physics ruminations that won't leave me alone...

Why do physicists consider higher-order (beyond our standard three) dimensions to be "rolled up" into a small space within the existing three dimensions?  They even spend a lot of time working out just how small this space must be given various theoretical and experimental results.  Yet if these additional dimensions are anything like the dimensions we know and love, I don't think there's any reason to think of them this way.  If they are orthogonal to the existing three dimensions, just extending in a direction we happen to be unable to perceive, then they needn't have any size at all in our existing set of dimensions.

I mean, if you look at a two-dimensional painting and then examine a line extending out of the plane of the painting into the third dimension, does it make any sense to think of that third dimension as somehow rolled up into the two dimensions of the painting?  No, it doesn't.  The third dimension is orthogonal to the other two dimensions.  It's kind of what orthogonal means--the new dimension does not align with the other two.  There is no overlap between the new and old dimensions.  Similarly, if the fourth spatial dimension is orthogonal to the first three, then I think it makes precisely as much sense to discuss its size in those original three dimensions as it actually takes up in those dimensions:  None at all.


  1. Perhaps in the beginning, right after the big bang, while 3 of the spatial dimensions grew to astronomical size, others were compactified; so the higher dimensions may be compact like a circle or ellipse with a very small size.

    We can see the 3 independent directions of 3D space by examining the corner of a room. Any other direction in 3D space is a linear combination of x,y and z, i.e; ax+by+cz, where a,b,c are any scalars. It’s not possible to orient a ruler perpendicular to all three edges at the corner of a room. If it is perpendicular to two edges, it would be parallel to the third. But if it could move in the fourth dimension we could get the ruler perpendicular to all three edges and meeting at a corner of a 3D room. The ruler would then be parallel to the 'fourth direction'. A particle that travels in 4D space, for example, can move along any linear combination of four independent directions. In addition to moving north/south & east/west, there is a fourth direction in the 4D space perpendicular to each of these other directions.

  2. Your first observation is how I think physicists currently imagine events to have unfolded (and folded :)). Your second observation is more in line with what I had in mind... Orthogonal should really mean orthogonal. If you tilt that ruler so it lies along the fourth dimension you might see only a cross section of it (since we can't perceive that fourth dimension), just the way the residents of Flatland would see only the outline of a ruler poking through their world and aligned with the third dimension, perpendicular to their world. Maybe I've read too much science fiction in general and Rudy Rucker in particular, but I don't have much trouble imagining additional dimensions. I even have what I think is a pretty unusual take on the idea that might be suitable for a science fiction story of my own, but who knows if it'll ever get written, sigh.