Chern Chuang, Bih-Yaw Jin, Wei-Chi Wei
Geometry is an essential ingredient of chemistry. The functionality of molecules depends heavily on their geometries. As chemists, we are interested in the connection of one type of molecules, the fullerenes composing of graphitic carbon, and its beaded counterpart. As one of the authors (BYJ) demonstrated in the Bridges conference last year, fullerene molecules are particularly suited to be models of making bead crafts. And the resulting beaded molecular models are both faithful representations of the molecules themselves and artistically appealing. Here we present two beaded molecules that correspond to the classic triply periodic minimal surfaces (TPMS), Schwarz's D and Schoen's G surfaces. These TPMSs are widely known to the chemical, material and mathematical societies, and are ubiquitous in Nature. By suitably inserting octagons into regular hexagonal honeycomb and introducing periodic boundary condition in all three dimensions, one obtains the fullerene correspondences of the TPMSs. In particular, octagons are represented by colored beads while hexagons are white in these beaded models.
The conjugate surface of the P surface. We chose to construct this surface in a tetrahedral form to avoid unconnected component. In contrast to the P surface, one can find this surface comprising helicoid units of two opposite chiralities, lining up along C2 axes. Octagonal rings are represented by green beads.
One of the most elusive embedded TPMSs, yet is found omnipresent in biological and material sciences. Along the line of decomposing P and D surfaces into connected catenoid and helicoid units, respectively, one can view this structure as connected helical surfaces that are half way along the catenoid-helicoid isometric deformation, and this beaded gyroid contains 16 such units each of the length of two translational units. In particular, we used three different colors for the eight-membered rings because these rings can be classified according to their face normal. In the usual right-handed Cartesian coordinates, eight-membered rings with face normal along x-axis are represented by purple beads, whereas ones along y-axis and z-axis are blue and green, respectively.