Dr. Alexey Ignatchenko Research Group

Replacement of hexagons by heptagons in the graphene honeycomb pattern creates a negative Gaussian curvature leading to the deviation (Fig. 1) from the graphene planar arrangement and induces ring closures. The ring closure can be done symmetrically in several ways which leads to different triply periodic minimal surfaces (TPMS). We have recently discovered new examples of the Gyroid, Schwartz P- and D-surfaces by the {3,7} pattern, known as the hyperbolic soccer ball. Click on each image to explore 3D interactive model. 
2 Layers of heptagons around the central one
Fig. 1. A beginning of the hyperbolic surface growth by adding layers of heptagons around the central one (shown in red). There are 7 heptagons in the first layer (grey), and 21 in the second layer (blue).
P-surface decorated by heptagons
Fig. 4. Schwartz P-surface made out of heptagons. Coloring shows that the same symmetry of heptagons tiling in P207-C168 structure can be visualized in several ways: four  pairs of adjacent trimers on the left side, or three pairs of circular tetramers on the right side.   

Interactive 3D model for the element of the Schwartz D-surface unit cell (requires javascript).
Controls: spin | no spin | spacefill | wire | ball&stick
labels on | off | color |
D-surface decorated by heptagons
Fig. 2. Element of the D-surface decorated by heptagons. It is constructed by joining four circular hexamers of heptagons.



TPMS