Fig. 2: Structure of the 600 cell.
From: Frustrated self-assembly of non-Euclidean crystals of nanoparticles
a Tetrahelices are chiral linear assemblies of tetrahedra. b Chiral ligands induce a twist between neighboring tetrahedra, and select and handedness of the tetrahelix. When the tetrahelix and the relative twist have the same (opposite) handedness, the distance between 4th-nearest neighbor tetrahedra’s centers d(1, 4) increases (reduces), thus lowering (increasing) their electrostatic repulsion. At the same time, the edges of the tetrahelix become more (less) coiled. c In 3D Euclidean space, two tetrahelices cannot fit side-to-side. d Two tetrahelices fitting perfectly together on the 3-sphere S3 (a stereographic projection). e The base icosahedron of the 600 cell. The 4 toroidal shells are indicated by color gradient, with the α coordinate of the boundaries between them marked. f–g 4 toroidal shells of the 600 cell, each containing 5 tetrahelices. f North pole bundle (0 < α < αN): 5 tetrahelices around the North pole of the base. g North ring of 5 tetrahelices wrapped around the North pole bundle (the latter is not visible). h South ring of 5 tetrahelices wrapped around the North ring. The surface triangles are facing the South pole of the base. The north and south rings together occupy the space αN < α < αS. i South pole bundle (αS < α < π/2): 5 tetrahelices are closely packed around the South pole of the base. j The coordinate Φμ = (α, β, θ) (stereographic projection). k The slice \(\bar{M}\), which contains the north and south rings (g and h, a total of 300 tetrahedra arranged in 10 tetrahelices, corresponding to the middle 10 triangles in the base icosahedron in e), are characterized by layers of constant α surfaces within αN < α < αS. The angle between (θ, β) evolves across the layers, according to the reference metric [Eq. (4)]. l The middle layer of \(\bar{M}\) is the Clifford torus (α = π/4, represented as a square with opposite edges identified). The θ axis is along the RH tetrahelices, and along the long side of the ribbon when RH tetrahelices assemble. The assembled ribbon (gray rectangle) has reference metric \(\bar{{{{{{{{\bf{g}}}}}}}}}\) from the 600 cell but is not limited by the size of the Clifford torus.
