I notice that each generation of matter is associated with a Lie group with that many degrees of freedom. Does that imply that 3 generations of matter is a direct result of 3 spatial dimensions?
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2$\begingroup$ This is numerology. Similar question: Why are there 4 Dimensions and 4 Fundamental Forces? $\endgroup$– Qmechanic ♦Commented Jan 2, 2018 at 15:38
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Right now we simply don't know why there are three generations of particles.
There are no theories (well, no credible theories) linking it to the dimensionality of spacetime. The closest we have is in String Theory where the number of dimensions is related to the topology of the Calabi-Yau manifold on which the theory is compactified. Whether this proves to be the correct explanation remains to be seen.
answered Jan 2, 2018 at 16:35