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OK, so I'm wondering how fast would the quarter have to travel to actually shrink to the size of a dime, based on special relativity?

The FitzGerald contraction would only shrink it in one dimension.

What would happen if the quarter was rotated on its disk at relativistic speeds?

Which is the best I could find for the question of shortening a quarter to the size of a dime using relativity.

The best "What if?"s always come from science geeks, I swear.