### Concept: Octahedron

#### 34

##### Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses

- Proceedings of the National Academy of Sciences of the United States of America
- Published over 6 years ago
- Discuss

The three known classes of convex polyhedron with equal edge lengths and polyhedral symmetry–tetrahedral, octahedral, and icosahedral–are the 5 Platonic polyhedra, the 13 Archimedean polyhedra–including the truncated icosahedron or soccer ball–and the 2 rhombic polyhedra reported by Johannes Kepler in 1611. (Some carbon fullerenes, inorganic cages, icosahedral viruses, geodesic structures, and protein complexes resemble these fundamental shapes.) Here we add a fourth class, “Goldberg polyhedra,” which are also convex and equilateral. We begin by decorating each of the triangular facets of a tetrahedron, an octahedron, or an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” We obtain the unique set of internal angles in each planar face of each polyhedron by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, and the variables are a subset of the internal angles in 6gons. Like the faces in Kepler’s rhombic polyhedra, the 6gon faces in Goldberg polyhedra are equilateral and planar but not equiangular. We show that there is just a single tetrahedral Goldberg polyhedron, a single octahedral one, and a systematic, countable infinity of icosahedral ones, one for each Goldberg triangle. Unlike carbon fullerenes and faceted viruses, the icosahedral Goldberg polyhedra are nearly spherical. The reasoning and techniques presented here will enable discovery of still more classes of convex equilateral polyhedra with polyhedral symmetry.

#### 27

##### Morphology Control of Anatase TiO2 Spindly Octahedra with Exposed High-Index {401} Facets and Application in Lithium-Ion Batteries

- Chemistry, an Asian journal
- Published over 7 years ago
- Discuss

Watch your figure: Three different shapes of TiO2 nanoparticles with various exposed facets have been synthesized by changing the adsorbents. The spindly octahedron with exposed high-index {401} facet exhibit the highest electrochemical performance than that of the other shapes.

#### 24

##### Discrete {Ni40} Coordination Cage: A Calixarene-Based Johnson-Type (J17) Hexadecahedron

- Journal of the American Chemical Society
- Published over 4 years ago
- Discuss

We report a Johnson hexadecahedronal coordination cage, constructed via ten Ni4-p-tert-butylthiacalix[4]arene (Ni4-TC4A) units as vertices and sixteen 5-(pyridin-4-yl)isophthalate (PIP) ligands as tiles. It features a gyroelongated square bipyramidal geometry, equivalent to two square pyramids pillared by a square antiprism, a J17 Johnson solid. Remarkably, the cage compound exhibits a much higher uptake capacity of C3H8 than CH4, representing a promising material for separation of these two gases. In contrast, Co4-TC4A units are linked by PIP ligands and rare {Co4O4Cl2} clusters, providing a one-dimensional bamboo stick-like polymer.

#### 4

##### Polyhedra Self-Assembled from DNA Tripods and Characterized with 3D DNA-PAINT

- Science (New York, N.Y.)
- Published over 6 years ago
- Discuss

DNA self-assembly has produced diverse synthetic three-dimensional polyhedra. These structures typically have a molecular weight no greater than 5 megadaltons (MD). We report a simple, general strategy for one-step self-assembly of wireframe DNA polyhedra that are more massive than most previous structures. A stiff three-arm-junction DNA origami tile motif with precisely controlled angles and arm lengths was used for hierarchical assembly of polyhedra. We experimentally constructed a tetrahedron (20 MD), a triangular prism (30 MD), a cube (40 MD), a pentagonal prism (50 MD), and a hexagonal prism (60 MD) with edge widths of 100 nanometers. The structures were visualized by transmission electron microscopy and by three-dimensional DNA-PAINT super-resolution fluorescent microscopy of single molecules in solution.

#### 3

##### Prescribed nanoparticle cluster architectures and low-dimensional arrays built using octahedral DNA origami frames

- Nature nanotechnology
- Published over 5 years ago
- Discuss

Three-dimensional mesoscale clusters that are formed from nanoparticles spatially arranged in pre-determined positions can be thought of as mesoscale analogues of molecules. These nanoparticle architectures could offer tailored properties due to collective effects, but developing a general platform for fabricating such clusters is a significant challenge. Here, we report a strategy for assembling three-dimensional nanoparticle clusters that uses a molecular frame designed with encoded vertices for particle placement. The frame is a DNA origami octahedron and can be used to fabricate clusters with various symmetries and particle compositions. Cryo-electron microscopy is used to uncover the structure of the DNA frame and to reveal that the nanoparticles are spatially coordinated in the prescribed manner. We show that the DNA frame and one set of nanoparticles can be used to create nanoclusters with different chiroptical activities. We also show that the octahedra can serve as programmable interparticle linkers, allowing one- and two-dimensional arrays to be assembled with designed particle arrangements.

#### 3

##### NANOCATALYSTS. Platinum-based nanocages with subnanometer-thick walls and well-defined, controllable facets

- Science (New York, N.Y.)
- Published over 5 years ago
- Discuss

A cost-effective catalyst should have a high dispersion of the active atoms, together with a controllable surface structure for the optimization of activity, selectivity, or both. We fabricated nanocages by depositing a few atomic layers of platinum (Pt) as conformal shells on palladium (Pd) nanocrystals with well-defined facets and then etching away the Pd templates. Density functional theory calculations suggest that the etching is initiated via a mechanism that involves the formation of vacancies through the removal of Pd atoms incorporated into the outermost layer during the deposition of Pt. With the use of Pd nanoscale cubes and octahedra as templates, we obtained Pt cubic and octahedral nanocages enclosed by {100} and {111} facets, respectively, which exhibited distinctive catalytic activities toward oxygen reduction.

#### 3

##### Computational design of self-assembling protein nanomaterials with atomic level accuracy.

- Science (New York, N.Y.)
- Published over 8 years ago
- Discuss

We describe a general computational method for designing proteins that self-assemble to a desired symmetric architecture. Protein building blocks are docked together symmetrically to identify complementary packing arrangements, and low-energy protein-protein interfaces are then designed between the building blocks in order to drive self-assembly. We used trimeric protein building blocks to design a 24-subunit, 13-nm diameter complex with octahedral symmetry and a 12-subunit, 11-nm diameter complex with tetrahedral symmetry. The designed proteins assembled to the desired oligomeric states in solution, and the crystal structures of the complexes revealed that the resulting materials closely match the design models. The method can be used to design a wide variety of self-assembling protein nanomaterials.

#### 2

##### Assembled molecular face-rotating polyhedra to transfer chirality from two to three dimensions

- Nature communications
- Published about 4 years ago
- Discuss

In nature, protein subunits on the capsids of many icosahedral viruses form rotational patterns, and mathematicians also incorporate asymmetric patterns into faces of polyhedra. Chemists have constructed molecular polyhedra with vacant or highly symmetric faces, but very little is known about constructing polyhedra with asymmetric faces. Here we report a strategy to embellish a C3h truxene unit with rotational patterns into the faces of an octahedron, forming chiral octahedra that exhibit the largest molar ellipticity ever reported, to the best of our knowledge. The directionalities of the facial rotations can be controlled by vertices to achieve identical rotational directionality on each face, resembling the homo-directionality of virus capsids. Investigations of the kinetics and mechanism reveal that non-covalent interaction among the faces is essential to the facial homo-directionality.

#### 1

##### Precise Molecular Fission and Fusion: Quantitative Self-Assembly and Chemistry of a Metallo-Cuboctahedron

- Angewandte Chemie (International ed. in English)
- Published over 5 years ago
- Discuss

Inspiration for molecular design and construction can be derived from mathematically based structures. In the quest for new materials, the adaptation of new building blocks can lead to unexpected results. Towards these ends, the quantitative single-step self-assembly of a shape-persistent, Archimedean-based building block, which generates the largest molecular sphere (a cuboctahedron) that has been unequivocally characterized by synchrotron X-ray analysis, is described. The unique properties of this new construct give rise to a dilution-based transformation into two identical spheres (octahedra) each possessing one half of the molecular weight of the parent structure; concentration of this octahedron reconstitutes the original cuboctahedron. These chemical phenomena are reminiscent of biological fission and fusion processes. The large 6 nm cage structure was further analyzed by 1D and 2D NMR spectroscopy, mass spectrometry, and collision cross-section analysis. New routes to molecular encapsulation can be envisioned.

#### 0

##### Synthesis and crystal structure of solvent-free dodecahydro closo-dodecaborate of nickel, NiB12H12

- Dalton transactions (Cambridge, England : 2003)
- Published over 2 years ago
- Discuss

Nickel closo-dodecaborate NiB12H12 was prepared by mechanosynthesis (ball milling) of mixtures of Na2B12H12 + NiCl2 followed by hydration and drying under dynamic vacuum. The crystal structures of hydrated and anhydrous closo-dodecaborates were characterized by temperature dependent synchrotron radiation X-ray powder diffraction, ab initio calculations, thermal analysis and infrared spectroscopy. Three different water containing complexes were found: a homoleptic octahedral complex in Ni(H2O)6B12H12 crystallizing in two different deformation variants of a complex centred closo-dodecaborate cube, and a heteroleptic octahedral complex in Ni(H2O)4B12H12 containing four water molecules and two hydrogens and centring also a deformed closo-dodecaborate cube. Anhydrous nickel closo-dodecaborate was obtained by drying the hydrated sample under dynamic vacuum. It crystallizes with bcc packing of B12H122- anions and Ni2+ is disordered close to the triangular face of the tetrahedral interstice coordinated by a H5 square pyramid.