Wikipedia Talk:Penrose tiling

use of the word "uncountable"

Sorry, Alfe, you're mistaken.  You said you branch a finite number of
times, but that only tiles a finite part of the plane.  If you want to
tile the entire plane, then you have to branch infinitely many times,
and that makes the number of tilings uncountable.  If we take them
modulo rotations and translations (placing a vertex at the origin and
including a horizontal edge) that's still uncountable.  So there are
uncountably many different tilings of the plane, but you only can tell
if you look at the whole plane.  If you look at a finite patch, all
the tilings agree (and each agrees in a set translation that is not
only infinite, but has positive density.) Only two of the tilings (up
to symmetry) have a fivefold center of symmetry.  -Dan Hoey 14:59, 5
June 2007 (UTC)