Question: Is this a coincidence? Or is it the main idea why physicists began to consider the possibility that the universe might be ten-dimensional?
No rigorous justification of your answer is necessary for me, since I don't think I have sufficient background to understand it. Answers linking to papers or citing references might be better received by other people who view this question and actually understand the answers, however.
Also I know that M theory is considered more up-to-date than string theory and it predicts an 11-dimensional universe (or at least one dimension higher), but my understanding was that M theory is a unifying framework for competing string theories, so basically one needs to understand string theory to understand M theory, I don't even understand string theory, so I don't want to worry about that right now. A yes/no answer to the ten-dimensional coincidence would be more than adequate for me right now. Perhaps keep in mind that other viewers will have different criteria.
Background/context: I was thinking about how the fact that O(3) has two path components actually corresponds to real-life geometric observations about rotations, reflections, and orientation. Then I thought "too bad it is a subset of 9-dimensional space and thus doesn't have any physical manifestation". Which led to the thought -- maybe it does, since string theory predicts that we live in more than three dimensions.
(It turns out this line of thinking was dumb anyway though since apparently SO(3) and presumably also O(3) is a three-dimensional manifold, so it actually wouldn't that be far-out to imagine that we lived in the tangent space of it, but whatever.)
Anyway I remembered that SO(3) wouldn't be the appropriate model to consider, since physical theories postulate that we actually live in four-dimensional spacetime, rather than three-dimensional spacetime, so I should look for the dimension of the analogous matrix group for "rotations" or "symmetries" for spacetime. I thought it was the Lorentz group (which doesn't work because it only has six dimensions) but Wikipedia corrected me and it turns out it is the Poincare group, which apparently is ten-dimensional -- just like string theories postulate for the universe.