"The forbidden crystal: Penrose tiling with molecules."

                Penrose tiling, a quasicrystal, as defined by Seong Joo Nam's research poster in University of Auckland Research Repository, Research Space as simplest version of 2-dimensional quasi-periodic tiling that comprises pentagonal motifs. Penrose tiling, an intriguing topic for first-time readers of science article since terms used is broad.  But then, Penrose tiling is a good example of tilings of the plane. A tiling is a covering of the plane, probably a floor by tiles with no overlaps and gaps. Gardner defines the tilings which have a finite number of shapes, called prototiles. There are known tilings (e.g., by squares or triangles) which are periodic; the translation of tiles in a specific distance and direction. If a tiling has no periods, it is said to be non-periodic which has underlying range order. Penrose tiling inspired by the idea of the mathematician and physicist Roger Penrose, investigated the set in which may be constructed to exhibit reflection symmetry and fivefold rotational symmetry. It has remarkable properties and notably accepted in the science community. First, the Penrose tiling classified as quasicrystals, are structural forms that are ordered but non-periodic. It produces Bragg diffraction and reveals both the fivefold symmetry. Second, it is self-similar in which same patterns occur at larger and larger scales. Hence, the tiling is obtained, and every patch shows that the tiling occurs infinitely many times.

The development of the Penrose tiling had evolved for years. When Roger Penrose introduced the tiling in his 1974 paper, it is composed of set six prototiles (P1) but based on pentagons rather than squares. There were gaps between the plane with regular pentagons, but Johannes Kepler showed that these gaps is filled with pentagrams (star polygons), decagon and related shapes. From six prototiles introduced by Penrose, subsequent discoveries reduces the number of prototiles of the tilings, the kite and dart tiling (P2) and the rhombus tiling (P3). The three types of Penrose tiling, P1-P3 share common features: The tiles are constructed from shapes related to pentagons and supplemented by matching rules to show non-periodic form. The Penrose tiling can be modified to alternately obtain a non-periodic of set of prototiles. The modification requires the label of vertices or edges and patterns of tile faces. The aesthetic value of tiling has been a long used for beautification of floors and ground. It has attracted attention and remains a source of interest to its visual appearance. The Penrose tiling is one of the decorative patterns in modern art of the Middle East. The collaboration of physicists, Peter J. Lu and Paul Steinhardt obtain evidences that Penrose tiling underlies examples depicting Islamic geometric patterns, such as tilings at the Darb-e Imam shrine in Isfahan.

The molecular approach of Penrose tilings, periodicity, can occur only for certain rotation symmetries: One-, Two-, Three-, Four and Six-fold symmetry axes are allowed; but Five-, Seven-, Eight- or higher-fold symmetry axes are considered forbidden (Lima-de-Faria, 1990). A new kind of solid was discovered in the year 1984 by Levin and Steinhardt which hypothetically violates the symmetry rules. Researchers such as Dan Shechtman, Denis Gratias, Ilan Belch and John Cahn announced the discovery of the solid which composes an Al-Mn alloy and its ability to diffract electrons like a crystal solid but  limited to icosahedral symmetry. Icosahedral symmetry includes six independent five-fold symmetry axes which are super forbidden. The quasicrystals were a new class of solids and could evade conventional crystallographic rules. The discovery of this solid depicts an atomic pattern and instead of being periodic, the researchers described it as the sum of two or more disproportionate periodic functions (L. Bindi et al., 2014).

Levine and Steinhardt showed that the five-fold symmetry is possible and classified as Penrose tiling since the tiles repeat with relatively unequal frequencies. The quasiperiodic principle of the quasicrystal is used to construct polyhedral units, protrusions, and holes on the faces that restrict the combination of polyhedral unit and joined together in a three-dimensional solid in an icosahedral symmetry manner. The discovery of quasicrystals had been one of the intriguing discoveries that even critics suggest the search for natural quasicrystals. Since then, almost hundred different types of quasicrystals is synthesised in the laboratory under carefully controlled directions. When Shechtman et al. (1984) published a pattern of diffraction images of quasicrystals, Levine and Steinhardt showed that the principle of quasicrystal is not a prediction but an evidence for an existing icosahedral quasicrystal. It gained criticism from the scientific community in which the assumption of atoms could not systematize into a fine pattern without introducing a high density of defects. The example presented by Shechtman et al. was highly defective; hence, the diffraction pattern does not conform to quasicrystal prediction.

In 1987, researchers An-Pang Tsai and collaborators reported the successful synthesis of the Al₆₃Cu₂₄Fe₁₃ quasicrystal (the synthetic analogue of natural icosahedrite). The synthesized icosahedrite is considered as bona fide quasicrystal. The discovery of this icosahedrite found as museum sample consisting of several typical rock-forming minerals with rare metal alloy minerals that is synthesised with khatyrkite and cupalite (L. Bindi et al., 2014). The Al₆₃Cu₂₄Fe₁₃ quasicrystal considered as the first naturally occurring quasicrystal even discovered in decades for the search of the forbidden crystal. Researchers suggest that quasicrystals are robust as crystals and have existed long before they it was synthesized in the laboratory. Moreover, the discovery would open new chapters of research in the field of mineralogy. The discovery of these icosahedrite would uplift business relating to mining or industries for the production of ores. New perspective plans will grow to obtain possible utilisation of materials used for construction of roads, cemented buildings, or even make metal alloys that are continuously undergo a long process of research. Economic interests will allow capitalists to invest in industries that study quasicrystals. Since these quasicrystals were promising to the future of the economy, the people behind these research increases numbers of workers and open opportunities to countries that continue to develop their economic status and enhance stability.