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As explained here, there is an infinite number of distinct tilings that can be constructed using the three sets of tiles introduced by Roger Penrose (rhomb, kite-dart, boat-star). The distinction between tiles and tilings is often not made, and one informally speaks of three types of Penrose tilings. Quasicrystals are real physical objects, without any mathematical connection to Penrose tilings. I don't know of any naturally ocurring quasicrystal that is well approximated by a Penrose tiling. Physicists would call a Penrose tiling a "toy model" of a quasicrystal.

As explained here, there is an infinite number of distinct tilings that can be constructed using the three sets of tiles introduced by Roger Penrose (rhomb, kite-dart, boat-star). The distinction between tiles and tilings is often not made, and one informally speaks of three types of Penrose tilings. 

Quasicrystals are real physical objects, without any mathematical connection to Penrose tilings. I don't know of any naturally ocurring quasicrystal that is accurately approximated by a Penrose tiling. Physicists would call a Penrose tiling a "toy model" of a quasicrystal. If you are interested in how quasicrystals can be modeled by a Penrose tiling, you might want to read Mackay's paper on What has the Penrose tiling to do with the icosahedral phases?

As explained here, there is an infinite set of tilings that can be constructed using the rhombic tiles introduced by Roger Penrose. So the term "Penrose tiling" is best understood as an informal term, and it would be more precise to speak of "Penrose tiles" (which come in three varieties, rhomb, kite-dart, boat star). Quasicrystals are real physical objects, without any mathematical connection to Penrose tilings. I don't know of any naturally ocurring quasicrystal that is well approximated by a Penrose tiling. Physicists would call a Penrose tiling a "toy model" of a quasicrystal.

As explained here, there is an infinite number of distinct tilings that can be constructed using the three sets of tiles introduced by Roger Penrose (rhomb, kite-dart, boat-star). The distinction between tiles and tilings is often not made, and one informally speaks of three types of Penrose tilings. Quasicrystals are real physical objects, without any mathematical connection to Penrose tilings. I don't know of any naturally ocurring quasicrystal that is well approximated by a Penrose tiling. Physicists would call a Penrose tiling a "toy model" of a quasicrystal.

As explained here, there is an infinite set of tilings that can be constructed using the rhombic tiles introduced by Roger Penrose. So the term "Penrose tiling" is best understood as an informal term, without a precise mathematical content. Quasicrystals are real physical objects, without any mathematical connection to Penrose tilings. I don't know of any naturally ocurring quasicrystal that is well approximated by a Penrose tiling. Physicists would call a Penrose tiling a "toy model" of a quasicrystal.

As explained here, there is an infinite set of tilings that can be constructed using the rhombic tiles introduced by Roger Penrose. So the term "Penrose tiling" is best understood as an informal term, and it would be more precise to speak of "Penrose tiles" (which come in three varieties, rhomb, kite-dart, boat star). Quasicrystals are real physical objects, without any mathematical connection to Penrose tilings. I don't know of any naturally ocurring quasicrystal that is well approximated by a Penrose tiling. Physicists would call a Penrose tiling a "toy model" of a quasicrystal.