Listen To A Shape Polyhedral Apartment

A Near-Miss Solid of Golden Joinery

RESIDENTS:A near-miss solid of golden joinery /

“An ultra-stable gold coordinated protein cage displaying reversible assembly”

Malay et al., Nature 569: 438-442 (2019)

An elegant near-miss solid was created at a molecular level. Hendecagonal ring-shaped protein complexes (concentric 11-mer complex of bacterial proteins, TRAPs, with amino acid substitutions) together with cross-linking gold ions assembled into a geometric supramolecular structure. It’s like a cage made of 24 hendecagons leaving big and small apertures on its surface. This shape is novel and also paradoxical. It is because, other than prisms and antiprisms, hendecagons are excluded from regular-faced polyhedra. The paradox was unraveled by geometrical insight. By connecting the center of each hendecagonal face (making a dual form), there emerged a snub cube, an Archimedean solid. Since the dual of the dual is the original polyhedron, the dual of the snub cube (pentagonal icositetrahedron) should represent this paradoxical structure. However, the pentagonal icositetrahedron has 24 faces of an irregular pentagon, but not the hendecagon. Authors figured out that nearly the same irregular pentagon is obtained by extending every other edge for 5 edges of the hendecagon, where differences of average edge lengths or included angles from the ideal shape fall within 0.5 % deviation! What a coincidence. A tiny geometrical discrepancy was completely disappeared within the molecular structure. The 5 edges of the hendecagon contributed to ring-ring interaction while other non-contributing 6 edges created big and small apertures between rings. By the way, the aim of this paper was a bit different, which opened an avenue to make a very stable supramolecular protein complex held by metal ions that can be readily disassembled into pieces by adding reducing reagents.
ROOM No.:V24[524]F38[33246]

Snub cube

One of the oddest Archimedean solids. It has 24 vertices (5-deg x24) and 38 faces (3-gon x32 and 4-gon x6). Octahedral symmetry with chirality. Each vertex is uniform, surrounded by five faces of four triangles and a square {3, 3, 3, 3, 4}. Making a space between squares of a cube followed by evenly filling the gap with 32 triangles results in a twist of squares either to the right or to the left. The isomorphic shape is obtained by dissecting each square (those sandwiched between triangles, 12 in total) into two triangles in the rhombic dodecahedron V24[424]F26[38418] or by dissecting each hexagon into four triangles in the truncated octahedron V24[324]F14[4668]. The dual figure is a pentagonal icositetrahedron V38[33246]F24[524], one of the Catalan solids with chirality.

Polyhedral Apartment

Have you ever visit an apartment of polyhedra?...


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