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### Nature Abhors Quasicrystals

January 16, 2012 The idiom that "Nature abhors a vacuum" is attributed to Baruch Spinoza, who developed a lot of interesting ideas that were certainly well ahead of his seventeenth century milieu. Scientists have the vacuum under control, both experimentally and theoretically, but what nature really abhors is crystals with five-fold symmetry. The reason for this is that space can't be filled with dodecahedrons, which are regular polyhedra with pentagonal sides. Johannes Kepler, in his book, Harmonices Mundi, showed that even when you drop down a dimension, pentagons are anathema. You can't fill a plane with regular pentagons, although irregular pentagons will do the job. The dodecahedron is one of the five Platonic solids, which are those solids that can be formed using the same regular polygon as faces. The others are the tetrahedron, cube, octahedron and icosahedron. Aristotle, in his book, On the Heavens, stated that the tetrahedron can fill space, but he was wrong.*Platonic solid sculptures in Steinfurt, Germany. (Photo by Zumthie, via Wikimedia Commons).*

The only Platonic solid that can fill space is the cube. If we relax our polyhedron requirement to solids that are formed through combinations of any regular polyhedra, not just one type, we get the Archimedean solids. Four objects in this class join the cube as space filling polyhedra. These are the triangular prism, hexagonal prism, truncated octahedron and gyrobifastigium.[1] Gyrobifastigium? I thought that mathematicians only drank coffee, but that's more like a beer pong word. When atoms arrange themselves in crystals, they necessarily fill space, and they do this in the simplest way possible by ordering themselves into small polyhedral cells that repeat. Not surprisingly, the majority of crystal cells are cubic and hexagonal crystals. Looking at the first series of transition metals, scandium through zinc, we have four hexagonal crystals, four body-centered cubic crystals, and two face-centered cubic crystals. As I summarized in a previous article (The 2011 Nobel Prize in Chemistry, October 7, 2011), no one believed Dan Shechtman in 1982 when he found five-fold symmetry in an alloy of aluminum that contained fourteen atomic percent manganese.[2-3] The material was novel, since it was formed by rapid solidification, a technique that inhibits motion of atoms as the material solidifies. Shechtman's electron diffraction data indicated an icosahedral point group symmetry. Icosahedra can't fill space. Not surprisingly, it took two years for Shechtman to get his paper accepted for publication; but after Shechtman's publication, other scientists came forward with examples of the same thing. In 1992 the International Union of Crystallography changed its definition of a crystal from "a regularly ordered, repeating three-dimensional pattern" to a solid with a "discrete diffraction diagram."[4] Shechtman was vindicated in his award of the 2011 Nobel Prize in Chemistry for his discovery of these quasicrystals, crystals that are ordered, but not periodic.

*A rather pretty tiling with five-fold rotational symmetry. This is a Penrose tiling, named after the mathematicial physicist, Roger Penrose.Penrose was issued a patent on his tiling in 1979.[5](Via Wikimedia Commons))*

The unusual way that quasicrystalline materials were synthesized in the laboratory seemed to indicate that they would be impossible to find in nature. Then, in 2009, a team of scientists from University of Florence (Florence, Italy), Princeton University and Harvard University, reported on the discovery of micrometer-sized quasicrystalline grains of an alloy of aluminum, copper, and iron in a mineral specimen from the Koryak Mountains in Siberia. It was not apparent how these crystals could have formed in nature.[6] Members of that same research team have published a paper in the Proceedings of the National Academy of Sciences that combines a little detective work with their usual science.[7-9] The detective work was because the provenance of the mineral specimen was not that well established. It was in the collection of a Florence museum, which bought it in 1990 from a private collector in Amsterdam, along with thousands of others.[9] The label on the specimen's box stated that the specimen came from the Koryak Mountains. Investigation found that the specimen was presumably smuggled from Russia through Romania. It was finally traced back to its originator, who had dug it from clay in a region of Russia at the Bering Strait.[9] Theoretical physicist, Paul Steinhardt of Princeton University, and his colleagues have visited that region looking for more quasicrystal specimens, but the results of that trek have yet to be published.[9] The small quasicrystals in the specimen are an icosahedrite phase of Al

_{63}Cu

_{24}Fe

_{13}that appear to have come from a meteorite.[7] These crystals are in combination with many other mineral phases, including a grain of stishovite, which is a phase of silica that is formed only at very high pressures (≥10 Gpa), likely through an impact.[7] Analysis of the oxygen isotope concentrations also show a non-terrestrial origin like that of carbonaceous chondrites.[8]. As the abstract of the PNAS paper summarizes,

"...Our evidence indicates that quasicrystals can form naturally under astrophysical conditions and remain stable over cosmic timescales, giving unique insights on their existence in nature and stability."[7]

### References:

- Space-Filling Polyhedron on Mathworld.
- D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, "Metallic phase with long range orientational order and no translation symmetry,", Physical Review Letters, vol. 53, no. 20 (November 12, 1984), pp. 1951-1953.
- D. Levine and R. Steinhardt, "Quasicrystals: a new class of ordered structures," Physical Review Letters, vol. 53, no. 26 (December 24, 1984), pp. 2477-2480.
- Kenneth Chan, "Israeli Scientist Wins Nobel Prize for Chemistry," The New York Times, October 5, 2011.
- Roger Penrose, "Set of tiles for covering a surface," US Patent No. 4,133,152, January 9, 1979.
- Luca Bindi, Paul J. Steinhardt, Nan Yao and Peter J. Lu, "Natural Quasicrystals," Science, vol. 324, no. 5932 (June 5, 2009), pp. 1306-1309.
- Luca Bindi, John M. Eiler, Yunbin Guan, Lincoln S. Hollister, Glenn MacPherson Paul J. Steinhardt and Nan Yao, "Evidence for the extraterrestrial origin of a natural quasicrystal," Proc. Natl. Acad. Sci., doi: 10.1073/pnas.1111115109, January 3, 2012.
- Impossible crystals are 'from space', BBC News January 3, 2012.
- Richard Van Noorden, "The quasicrystal from outer space," Nature News, January 3, 2012.

*Permanent Link to this article*

Linked Keywords: Idiom; Baruch Spinoza; seventeenth century; vacuum; ultra-high vacuum; vacuum state; crystal; five-fold symmetry; dodecahedron; regular polyhedra; pentagon; Johannes Kepler; Harmonices Mundi; dimension; irregular pentagon; Platonic solid; tetrahedron; cube; octahedron; icosahedron; Aristotle; On the Heavens; Steinfurt, Germany; Zumthie; Wikimedia Commons; Archimedean solids; triangular prism; hexagonal prism; truncated octahedron; gyrobifastigium; mathematicians only drank coffee; beer pong; cubic; scandium; zinc; hexagonal crystals; body-centered cubic; face-centered cubic; Dan Shechtman; alloy; aluminum; manganese; rapid solidification; atom; electron diffraction; icosahedral symmetry; icosahedral point group symmetry; International Union of Crystallography; Nobel Prize in Chemistry; quasicrystal; Penrose tiling; mathematicial physicist; Roger Penrose; University of Florence (Florence, Italy); Princeton University; Harvard University; micrometer; copper; iron; mineral; Koryak Mountains; Siberia; Proceedings of the National Academy of Sciences; provenance; Florence; Amsterdam; Russia; Romania; Bering Strait; theoretical physicist; Paul Steinhardt; meteorite; stishovite; silica; high pressure; Pascal; Gpa; oxygen isotope; carbonaceous chondrite; US Patent No. 4,133,152.

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