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The binary Cd5.7Yb phase representing the Tsai‐type category of the icosahedral quasicrystals is solved by the assignment of a unique atomic decoration to rhombohedral units in the Ammann–Kramer–Neri tiling. The unique decoration is found for units with an edge length of 24.1 Å and 3m internal point symmetry. The structural refinement was carried out for two underlying tilings generated by the projection method for 6D space. The difference lies in the location of the origin point which for one tiling is in the vertex and for the second one in the center of the 6D unit cell. The two tilings exhibit mutual duality. The choice of the tiling has a minor effect on the final structural model as both converge to an R factor of ∼11.5%. The main difference is related to the treatment of the Cd4 tetrahedral motif which is either orientationally ordered and aligned with the threefold axis or disordered and modeled as a partially occupied icosahedron. Both models can be presented as a covering by rhombic triacontahedral clusters with identical positions of clusters within rhombohedral units. The shell structure is Tsai‐type in the case of the first tiling and Bergman‐type for the other. [ABSTRACT FROM AUTHOR]

This paper is part of a series that describes the Fibonacci icosagrid quasicrystal (FIG) and its relation to the E 8 root lattice. The FIG was originally constructed to represent the intersection points of an icosahedrally symmetric collection of planar grids in three dimensions, with the grid spacing of each following a Fibonacci chain. It was found to be closely related to a five-fold compound of 3D sections taken from the 4D Elser–Sloane quasicrystal (ESQC), which is derived via a cut-and-project process from E 8 . More recently, a direct cut-and-project from E 8 has been found which yields the FIG (presented in another paper of this series). The present paper focuses not on the full quasicrystal, but on the relationship between the root polytope of E 8 (Gosset's 4 21 polytope) and the core polyhedron generated in the FIG, a compound of 20 tetrahedra referred to simply as a 20-Group. In particular, the H 3 symmetry of the FIG can be seen as a five-fold or "golden" composition of tetrahedral symmetry (referring to the characteristic appearance of the golden ratio). This is shown to mirror a connection between tetrahedral and five-fold symmetries present in the 4 21 . Indeed, the rotations that connect tetrahedra contained within the 4 21 are shown to induce, in a certain natural way, the tetrahedron orientations in the 20-Group. [ABSTRACT FROM AUTHOR]

A comprehensive critical survey of structures of exotic nano-, meso- and microdiamonds with dodecahedral and icosahedral symmetry (N/MDPS) is presented. Due to their high dodecahedral or icosahedral symmetry, the unique complex atomic and electronic structure of N/MDPS leads to transport and mechanical properties very promising for photonic, quantum, and nanomechanical applications. To explain the nature of diamonds, theoretical models have been proposed based on the formation of twinned structures consisting of either 5 or 20 symmetrically equivalent tetrahedral and prismatic fragments of the face-centered cubic lattice with the formation of star-shaped or icosahedral clusters, respectively. It has been shown that these twinned nano- and mesodiamonds have limited dimensions due to accumulation of uncompensated structural stresses arising from the deviation of the angles between diamond <111> facets from perfect 72° in tetrahedral fragments of the face-centered cubic lattice to 70.5° between five symmetrically equivalent twinned fragments. [ABSTRACT FROM AUTHOR]

Colloidal particles form a face‐centered‐cubic (FCC) lattice on crystallization; however, recent studies have indicated that the cluster with the icosahedral symmetry can be formed under spherical confinement. The icosahedral cluster appears only when the number of constituent particles is small because the FCC lattice is favored thermodynamically for a bulk colloidal crystal. This paper reports the observation of a large icosahedral colloidal cluster comprising ≈4.9 × 105 colloidal particles, which is considerably larger than the previously reported number. This large icosahedral cluster clearly exhibits the triangular reflection pattern that corresponds to one face of the icosahedron. Further, an oblique illumination with sample rotation is found to be an effective method to characterize the icosahedral cluster optically. In this study, larger colloidal particles are used than those in previous studies. It is hypothesized that slow reconfiguration during the evaporation process can result in the formation of a large icosahedral cluster due to the large particle size. [ABSTRACT FROM AUTHOR]

© 2015 Taylor & Francis Determining the structure of an icosahedral virus crystal by X-ray diffraction follows very much the same course as conventional protein crystallography. The major differences arise from the relatively large sizes of the particles, which significantly affect the data collection process, data processing and management, and later, the refinement of a model. Most of the other differences are due to the high 5 3 2 point group symmetry of icosahedral viruses. This alters dramatically the means by which initial phases are obtained by molecular substitution, extended to higher resolution by electron density averaging and density modification, and the refinement of the structure in the light of high non-crystallographic symmetry. In this review, we attempt to lead the investigator through the various steps involved in solving the structure of a virus crystal. These steps include the purification of viruses, their crystallization, the recording of X-ray diffraction data, and its reduction to structure amplitudes. It further addresses the problems attending phase determination and ultimately the refinement of a model. Finally, we describe the unique properties of virus crystals and the factors that influence their physical and diffraction properties.

© 2015 Taylor & Francis Determining the structure of an icosahedral virus crystal by X-ray diffraction follows very much the same course as conventional protein crystallography. The major differences arise from the relatively large sizes of the particles, which significantly affect the data collection process, data processing and management, and later, the refinement of a model. Most of the other differences are due to the high 5 3 2 point group symmetry of icosahedral viruses. This alters dramatically the means by which initial phases are obtained by molecular substitution, extended to higher resolution by electron density averaging and density modification, and the refinement of the structure in the light of high non-crystallographic symmetry. In this review, we attempt to lead the investigator through the various steps involved in solving the structure of a virus crystal. These steps include the purification of viruses, their crystallization, the recording of X-ray diffraction data, and its reduction to structure amplitudes. It further addresses the problems attending phase determination and ultimately the refinement of a model. Finally, we describe the unique properties of virus crystals and the factors that influence their physical and diffraction properties.

The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in P 5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group S 5 . We give canonical explicit S 5 -invariant Pfaffian equations through a 6 × 6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of S 5 . Finally, we give S 5 -invariant equations for the embedding of Y inside (P 1) 5 , and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4. [ABSTRACT FROM AUTHOR]

Satellite tobacco mosaic virus (STMV) is among the smallest viruses, having 60 identical subunits arranged with T = 1 icosahedral symmetry. Its crystal structure was solved at 290 K and was refined using, in part, crystals grown in microgravity. Electron-density maps revealed nearly 57% of the genomic ssRNA. Using six flash-cooled crystals, diffraction data were recorded to 1.4 Å resolution and independent refinements of the STMV model were carried out versus the previous 1.8 Å resolution data representing merged data from 21 crystals (271,689 unique reflections), data consisting of corresponding reflections to 1.8 Å resolution from the cooled crystals and 1.4 Å resolution data from the cooled crystals comprised of 570,721 unique reflections. Models were independently refined with full NCS constraints using the program CNS and in restrained mode using the programs CNS, REFMAC5 and SHELX-97, with the latter two procedures including anisotropic temperature factors. Significant additional structural detail emerged from the analyses, including a unique cation and anion arrangement on fivefold axes and a precise assessment of icosahedral symmetry exactness in the crystal lattice. STMV represents the highest resolution native virus structure currently known by a substantial margin, and it permits the evaluation of a precise atomic model of a spherical virus at near-atomic resolution for the first time.

In this paper we analyze nonlinear dynamics of the fullerene molecule. We prove, under isotypic nonresonant assumptions, the existence of global branches of periodic solutions emerging from an icosahedral equilibrium (nonlinear normal modes). We use equivariant degree theory to prove the existence of several branches of symmetric solutions for maximal orbit types which are standing and rotating waves that propagate along the molecule with icosahedral, tetrahedral, pentagonal, and triangular symmetries. Using a model that reflects the nonlinear characteristics of the fullerene, we confirm numerically the isotypic nonresonant assumptions and apply numerical Newton's continuation method to approximate some of the predicted modes. [ABSTRACT FROM AUTHOR]

In this paper we analyze nonlinear dynamics of the fullerene molecule. We prove, under isotypic nonresonant assumptions, the existence of global branches of periodic solutions emerging from an icosahedral equilibrium (nonlinear normal modes). We use equivariant degree theory to prove the existence of several branches of symmetric solutions for maximal orbit types which are standing and rotating waves that propagate along the molecule with icosahedral, tetrahedral, pentagonal, and triangular symmetries. Using a model that reflects the nonlinear characteristics of the fullerene, we confirm numerically the isotypic nonresonant assumptions and apply numerical Newton's continuation method to approximate some of the predicted modes. [ABSTRACT FROM AUTHOR]

The capsids of large, icosahedral dsDNA viruses are built from well-ordered aggregates of capsomers, known as trisymmetrons and pentasymmetrons, which are centered on the icosahedral 3-fold and 5-fold axes, respectively. We derive the complete set of rules for constructing an icosahedral structure from these symmetrons when the T lattice symmetry is odd and show that there are three classes of solutions, each of which follows from a different relationship between the size of the pentasymmetron and the values of the h and k icosahedral lattice parameters. Together, these three classes account for all possible ways of building an icosahedral structure solely from trisymmetrons and pentasymmetrons. Also, every icosahedral lattice with odd T number has solutions from exactly two of these three classes, with the set of allowed classes dependent on which of the two lattice parameters is odd. For symmetric lattices (if h=k or h=0), the two solutions yield the same symmetron sizes, but when the lattice parameters are equal (h=k) the solutions can be distinguished by the relative orientations of the symmetrons. We discuss these results in the context of known virus structures and explore the implications for viruses whose structures have not yet been solved.

Many viral capsids follow an icosahedral fullerene-like structure, creating a caged polyhedral arrangement built entirely from hexagons and pentagons. Viral capsids consist of capsid proteins,which group into clusters of six (hexamers) or five (pentamers). Although the number of hexamers per capsid varies depending on the capsid size, Caspar-Klug Theory dictates there are exactly twelve pentamers needed to form a closed capsid.However, for a significant number of viruses, including viruses of the Papovaviridae family, the theory doesn’t apply. The anomaly of the Caspar-Klug Theory has raised a new question:“For which Caspar and Klug models can each hexamer be replaced with a pentamer while still following icosahedral symmetry?” This paper proposes an answer to this question by examining icosahedral viral capsid-like structures composed only of pentamers, called pentagonal polyhedra. The analysis shows that pentagonal polyhedra fall in a subclass of T, defined by P ≥ 7 and T = 1( mod 3).

The atomic structure of medium-range order in metallic glasses is investigated by using molecular dynamics (MD) simulations. Glass formation processes were simulated by rapid cooling from liquid phases of a model binary alloy system of different-sized elements. Two types of short-range order of atomic clusters with the five-fold symmetry are found in glassy phases: icosahedral clusters (I-clusters) formed around the smaller-sized atoms and Frank–Kasper clusters (i.e., Z14, Z15, and Z16 clusters (Z-clusters)) formed around the bigger-sized atoms. Both types of clusters (I-and Z-clusters) are observed even in liquid phases and the population of them goes up as the temperature goes down. A considerable atomic size difference between alloying elements would enhance the formation of both the I- and Z-clusters. In glassy phases, the I- and Z-clusters are mutually connected to form a complicated network, and the network structure becomes denser as the structural relaxation goes on. In the network, the medium-range order is mainly constructed by the volume sharing type connection between I- and Z-clusters. Following Nelson’s disclination theory, the network structure can be understood as a random network of Z-clusters, which is complimentarily surrounded by another type of network formed by I-clusters.

The optical behavior of both synthetic and natural icosahedral and decagonal quasicrystals in the range 400–700 nm has been explored by repeated measurements of the reflectance values on crystallographically oriented fragments. We found differences in the optical behavior between natural and synthetic quasicrystals, and discuss the possible reasons to justify this feature. Natural quasicrystals, both icosahedral and decagonal, exhibit an almost identical chemical composition to their synthetic analogs, but seem to show a higher degree of structural perfection, as indicated from their electron and X-ray diffraction patterns. Such perfection is mainly due to the very low phason strain shown by the minerals with respect to the synthetic compounds and substantiates the differences in the measured reflectance values. The new finding constitutes significant new support for the original proposal that icosahedral and decagonal quasicrystals can be energetically stable states of matter, on the same footing as crystals. Electron diffraction pattern of icosahedrite (left) and decagonite (right). A high-resolution transmission electron image of icosahedrite is on the background. [ABSTRACT FROM AUTHOR]

The multiple-cell approach is discussed as an alternative to the higher-dimensional crystallographic description of icosahedral quasicrystals. Four types of quasi-unit cells fill the space without gaps and overlappings. Every cell in the whole tiling is decorated by specific atoms in a particular way and is associated with a triad: type, position, and orientation. The key features of the proposed approach are the subgroup/coset decomposition of icosahedral symmetry groups in accordance with the orbit-stabilizer theorem, a strict mathematical formalization of the substitution rules for all types of quasi-unit cells in the Socolar-Steinhardt tiling, formalization of the recursive inflation/deflation rules, and the eigenvalue-eigenvector analysis of corresponding substitution matrices. The similar approach can be applied to almost all types of quasicrystals. [ABSTRACT FROM AUTHOR]

An original structural model for the description of icosahedral quasicrystals is proposed. This novel theoretical approach is based on a surface-stacking image of icosahedral quasicrystals and is in concert with the step-terrace morphology observed in many experimental investigations of real icosahedral surfaces. According to this model, the infinite icosahedral structure is constructed by stacking two arrangements of quasi-periodic surfaces along each of the 5-fold axis of the icosahedral symmetry. In this stacking order, the spacings between the layered surfaces are governed by Fibonacci sequence. The proposed model operates within the real physical (Euclidean) space and provides a direct three-dimensional visual representation of the icosahedral superstructure, which can be instrumental for conducting different modes of experimentations, analyses, and fabrication strategies; eliminating a major roadblock for researchers especially outside the fields of crystallography and material science. More importantly, understanding the long-range surface-stacking logic of the icosahedral structure will hopefully provide a deeper understanding of the structure of quasicrystals at an atomic scale and help achieve improved control over material compositions and structure. [ABSTRACT FROM AUTHOR]

Three new crystal forms of satellite panicum mosaic virus (SPMV) were grown and their structures solved from X-ray diffraction data using molecular replacement techniques. The crystals were grown under conditions of pH and ionic strength that were appreciably different then those used for the original structure determination. In rhombohedral crystals grown at pH 8.5 and low ionic strength PEG 3350 solutions, Fourier syntheses revealed segments, ten amino acid residues long, of amino-terminal polypeptides not previously seen, as well as masses of electron density within concavities on the interior of the capsid, which appeared in the neighborhoods of icosahedral five- and threefold axes. The densities were compatible with secondary structural domains of RNA, and they included a segment of double helical RNA of about four to five base pairs oriented, at least approximately, along the fivefold axes. The distribution of RNA observed for SPMV appears to be distinctly different than the encapsidated nucleic acid conformation previously suggested for another satellite virus, satellite tobacco mosaic virus. This study further shows that analysis of viruses in crystals grown under different chemical conditions may reveal additional information regarding the structure of encapsidated RNA.

Abstract With the recent breakthroughs in experimental technologies, structure determination of extremely large assemblies, many with icosahedral symmetry, has been rapidly accelerating. Computational studies of their dynamics are important to deciphering their functions as well as to structural refinement but are challenged by their extremely large size, which ranges from hundreds of thousands to even millions of atoms. Group theory can be used to significantly speed up the normal mode computations of these symmetric complexes, but the derivation is often obscured by the complexity of group theory and consequently is not widely accessible. To address this problem, this work presents an easy recipe for normal mode computations of complexes with icosahedral symmetry. The recipe details how the Hessian matrix in symmetry coordinates can be constructed in a few easy steps of matrix multiplications, without going through the complexity of group theory. All the "ingredient" matrices required in the recipe are fully provided in the Supplemental Information for easy reproduction. The work is timely considering the expected large in-flux of many more icosahedral assemblies in the near future. The recipe uses a minimum amount of memory and solves the normal modes in a significantly reduced amount of time, making it feasible to perform normal mode computations of these assemblies on most computer systems. Graphical abstract Image 1 Highlights • Hessian matrix in symmetry coordinates constructed via simple linear combinations. • An easy recipe for efficient normal mode computations of icosahedral complexes. • All the ingredients matrices required are fully provided for easy reproduction. • Work is timely considering the expected large in-flux of many more such complexes. [ABSTRACT FROM AUTHOR]

The development of antiviral strategies requires a clear understanding of the principles that control the protein arrangements in viral shells. Considered here are those capsids that violate the paradigmatic Caspar and Klug (CK) model, and it is shown that the important structural features of such anomalous shells from the Picobirnaviridae, Flaviviridae and Leviviridae families can be revealed by models in the form of spherical icosahedral packings of equivalent rhombic structural units (SUs). These SUs are composed of protein dimers forming the investigated capsids which, as shown here, are based on the rhombic triacontahedron (RT) geometry. How to modify the original CK approach in order to make it compatible with the considered rhombic tessellations of a sphere is also discussed. Analogies between capsids self‐assembled from dimers and trimers are demonstrated. This analysis reveals the principles controlling the localization of receptor proteins (which recognize the host cell) on the capsid surface. The symmetry of the capsomers forming a viral shell determines the polyhedron underlying the shell structure. If the capsid is self‐assembled from dimers, this is the rhombic triacontahedron with 30 equivalent rhombic faces. [ABSTRACT FROM AUTHOR]

A multiple-cell approach is discussed as a possible alternative to the higher dimensional crystallography of icosahedral quasicrystals. It is based on the Socolar- Steinhardt tiling combined with the quasi-unit cell model. Quasi-unit cells fill the space without gaps and overlappings similar to those in periodic crystals. Similarly, the atoms can occupy general and special positions. The alloy stoichiometry and the packing density can be calculated through the relative tile frequencies, which in turn are determined as the components of the Perron-Frobenius eigenvector of the corresponding substitution matrix. The calculation of the diffraction pattern reduces to the Perron projection of a special matrix, the entries of which reflect the contribution of each type of quasi-unit cell to the coherent scattering. [ABSTRACT FROM AUTHOR]