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here's nureyev's tomb and no, it's not a rug tossed over it, that's it:

Which quickly brings to mind the question of what happens when the cloth deteriorates? Ewwwww.

It's probably concrete finished to look like cloth. Can't really tell from the photo though.

it's not cloth, he died in the early 90's, so it definitely would a bit worse for wear by now. even if nobody's walked on it.

apparently he was a rug collector.