James Propp: "I seem to recall hearing a strong version of this stated once, to the effect that if you have a pattern of diameter n in one Penrose tiling, and you look for that pattern in a second Penrose tiling, you won't have to travel further than distance 2n." Here's how it was stated in Martin Gardner's "Penrose Tiles to Trapdoor Ciphers" (1989): "Suppose you have explored a circular region of diameter d. Call it the 'town' where you live. Suddenly you are transported to a randomly chosen parallel Penrose world. How far are you from a circular region that exactly matches the streets of your home town? Conway answers with a truly remarkable theorem. The distance from the perimeter of the home town to the perimeter of the duplicate town is never more than d times half of the cube of the golden ratio, or 2.11+ times d. (This is an upper bound, not an average.) If you walk in the right direction, you need not go more than that distance to find yourself inside an exact copy of your home town. The theorem also applies to the universe in which you live. Every large circular pattern (there is an infinity of different ones) can be reached by walking a distance in some direction that is certainly less than about twice the diameter of the pattern and more likely about the same distance as the diameter." I should point out that this is an altered version of Gardner's original January 1977 column text where it is stated that "the distance is never more than 2d".