Quasicrystal

Conventional crystallography holds that crystals cannot have 5-fold rotational symmetry. The reason is geometrical: regular pentagons cannot tile a plane without gaps, so a repeating lattice - which by definition must fill space completely - cannot be built around a 5-fold axis. Quasicrystals break this rule. ^[1] They are ordered solids with genuine rotational symmetry at angles - including 5-fold, 7-fold, and 10-fold - that are geometrically impossible in any conventional crystal lattice. ^[1] The order is real: quasicrystals are not random or amorphous. What they lack is translational symmetry - the repeating-unit arrangement that defines a conventional crystal lattice.

Structure: Penrose Tiling

The internal structure of quasicrystals is equivalent to a three-dimensional version of Penrose tiling - an arrangement of differently shaped tiles that cover a surface completely and systematically, producing rotational symmetry, but without any repeating unit that translates to fill the whole area. ^[1] In a Penrose tiling with two differently shaped diamond tiles, 5-fold symmetry emerges from the arrangement rules even though no single patch of tiles repeats exactly. The pattern is not random - every placement is governed by matching rules - but you will never find one section of the tiling that simply slides (translates) onto another section and matches it perfectly.

This distinction is important. Conventional crystals have both rotational symmetry and translational symmetry: every unit cell is an exact copy of every other. Quasicrystals have rotational symmetry without translational symmetry. They occupy a genuinely new structural category - ordered but non-periodic.

Natural Quasicrystal Minerals

Two quasicrystal minerals have been identified in nature, both recovered from the Khatyrka meteorite found in southeastern Chukotka, Russia. ^[1] Their compositions are as follows.

| Mineral | Formula | |---|---| | Icosahedrite | Al₆₃Cu₂₄Fe₁₃ ^[1] | | Decagonite | Al₇₁Ni₂₄Fe₅ ^[1] |

Icosahedrite takes its name from the icosahedron - a polyhedron with 20 triangular faces and 5-fold symmetry axes - reflecting the fivefold symmetry of its atomic arrangement. Decagonite’s name refers to the decagon (10-sided polygon), indicating 10-fold symmetry. Both names encode the symmetry type that distinguishes quasicrystals from all conventional minerals.

The extraterrestrial origin of the Khatyrka meteorite is significant. It implies that quasicrystal formation can occur under the high-energy conditions of space - processes quite different from the geological environments that produce most terrestrial minerals. Whether additional natural quasicrystals exist, either in other meteorites or in terrestrial settings, remains an open question.

References

1. Nesse, W. D. (2018). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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