Listen To A Shape Polyhedral Apartment

Self-Assembled Beauty

RESIDENT:Self-Assembled Beauty

“Self-assembly of tetravalent Goldberg polyhedra from 144 small components” Fujita et al., Nature 540: 563-566 (2016)

Natural wonders may extend to a molecular phenomenon in which limited types of small components assemble into a highly ordered structure. In chemistry, geometric supramolecular structures were obtained by self-assembly of metal ions (M: as tetravalent vertices) and organic ligands (L: as bent edges at a certain angle). X-ray crystallography revealed that the complex M3060 made of 90 components has a structure resembling a tetravalent Goldberg polyhedron tet-G(2,1) typeIII, an unprecedented structure in the molecular world. The authors predicted even larger complexes, and successfully obtained M4896 made of 144 components (tet-G(2,2), type II). Self-assembled structures are thermodynamically stable. Geometric principles may underlie formation of supramolecular structures.
ROOM No.:V30[430]F32[38424]

tetravalent Goldberg polyhedra tet-G(2,1), or GC2,1(Octahedron)

tetravalent Goldberg polyhedra* V30[430]F32[38424]
Polyhedra obtained by the Goldberg-Coxeter construction* on an octahedron. Simply, these are created by adding tetragons to the octahedron, 4-reguraly and symmetrically (octahedral). Triangles are 8 in total. Triangles are located at vertices of a cubic cell, and tetragons fill the gaps evenly. Starting from one triangle, move m-faces over tetragons in one direction, turn 90 degrees, and move n-faces to reach the closest triangle. A pair of numbers (m,n) is useful to classify the structures: m, n=0 (type I), m=n (type II), and m≠n (type III). Only type III is chiral, and looks twisted. Although triangles and tetragons are both familiar, tetravalent Goldberg polyhedra look unusual yet so beautiful. Type II and III are particularly interesting.
* Please be advised that the terms “tetravalent Goldberg polyhedra” and “Goldberg-Coxeter construction” are used for convenience, but might not be accurate. See, Brinkmann et al., Proceedings Royal Society A 473: 20170267 (2017) Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations

Polyhedral Apartment

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