Does a plane exists in three dimensions? 

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Yes and no.Yes, because all 3D figures are actually made of planes, lines, points, or any other 1 and 2D figure.No, because, by itself, a plane doesn't exist in 3D space, because 3D space needs length, breadth and hight, whereas a plane only has length, and breadth.Hope that helps.Cheers.

no a plane has no thickness