page_title: Geometrical frustration
section_title: Regular packing of tetrahedra: the polytope {3,3,5}
hierarchical_section_title: Geometrical frustration / Geometric frustration without lattice / Regular packing of tetrahedra: the polytope {3,3,5}
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context_page_description: In condensed matter physics, the term geometrical frustration refers to a phenomenon, where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces lead to quite complex structures. As a consequence of the frustration in the geometry or in the forces, a plenitude of distinct ground states may result at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much studied examples are amorphous materials, glasses, or dilute magnets.
  The term frustration, in the context of magnetic systems, has been introduced by Gerard Toulouse. Indeed, frustrated magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor spins coupled antiferromagnetically, by G. H. Wannier, published in 1950. Related features occur in magnets with competing interactions, where both ferromagnetic as well as antiferromagnetic couplings between pairs of spins or magnetic moments are present, with the type of interaction depending on the separation distance of the spins.
context_section_description: Twenty irregular tetrahedra pack with a common vertex in such a way that the twelve outer vertices form a regular icosahedron. Indeed, the icosahedron edge length l is slightly longer than the circumsphere radius r (l ≈ 1.05r). There is a solution with regular tetrahedra if the space is not Euclidean, but spherical. It is the polytope {3,3,5}, using the Schläfli notation, also known as the 600-cell.
  There are one hundred and twenty vertices which all belong to the hypersphere S³ with radius equal to the golden ratio (φ = ) if the edges are of unit length. The six hundred cells are regular tetrahedra grouped by five around a common edge and by twenty around a common vertex. This structure is called a polytope (see Coxeter) which is the general name in higher dimension in the series containing polygons and polyhedra. Even if this structure is embedded in four dimensions, it has been considered as a three dimensional (curved) manifold. This point is conceptually important for the following reason. The ideal models that have been introduced in the curved Space are three dimensional curved templates. They look locally as three dimensional Euclidean models. So, the {3,3,5} polytope, which is a tiling by tetrahedra, provides a very dense atomic structure if atoms are located on its vertices. It is therefore naturally used as a template for amorphous metals, but one should not forget that it is at the price of successive idealizations.
caption: 600-cell: polytope {3,3,5}
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