Alternate Title: It's a point, dammit!

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.