Chuang made this beaded model in order to illustrate the connection between standard dodecahedral structures and two-layer high-genus fullerenes. He has intentionally pushed the vertices inward so one can see the possibility of hole opening at that position.
C168 is quite uniqe because it correponds to C60 in the hyperbolic space. We can view standard fullerenes as a tiling of graphene sheet on a sphere, which is a two-dimensional manifold with postive curvature everywhere. C60 corresponds to the smallest fullerene with all pentagons separated by only one CC bond. Similarly, C168 is the smallest fullerene in a hyperbolic space with all heptagons separated by only one CC bond.
In the bead model I created, purple beads stand for the edges of heptagons, and white beads are the CC bonds separating different heptagons.
This amazing beaded structure is connected by a tetravalent network, which is different our previous trivalent beaded fullerenes. The construction rule for this type of toroidal systems, however, is exactly similar to that we used for carbon tori. But, instead of pentagons and heptagons, here we use triangle and pentagons to simulate the positive and negative Gaussian curvatures located in the outer- and inner-rim of the torus. Generalization to other topologically nontrivial 2-D structures seems to be straightforward.
I try to experiment with the idea that using three beads to stand for an edge of polyhedron. In this case, I made a tetrahedron. Previously, I have posted some of this kind of models. The original idea came from Chuang again.
Here is an icosahedral fullerene in which the sharp vertices are forced innerward such that we can view this structure as the precursor for forming the outer part of a hight-genus fullerene. (created by Chuang)
In this proposal, we continue the project in academic year 2007 named ┌Chemistry, Geometry, and Art: The Beaded Molecules┘, in which we discussed the viability of handicraft beading to represent general fullerene molecules with all sorts of topologies and geometries. We will further focus on newly discovered zeolite/triply periodic minimal surfaces and, in particular, high genus structures, which scarcely appear in the literature. As we have learned in previous experience, the beaded molecules nicely simulate fullerenes with finite variations on their embedded geometrical object such as icosahedrons in the usual Ih-symmetric fullerenes. This is originated from the fact that the interplay between the hard-sphere interaction between adjacent beads and the tensile force exerted by the thread inevitably results in a relatively high recovery force constant, in other words, the beaded model has high mechanical stability. We believe that in the mean while students making beaded molecules will be acquainted with molecular geometry besides having a great deal of pleasure about the beauty of beaded molecules. Short-term workshop teaching the basic theory and techniques of beading beaded fullerene will be presented. Refinement and augmentation of currently established homepage and other internet resources will be made.
Yesterday I went to IKEA searching for an adequate book shelf, and I was happened to find a nice light stand (I don't know the word... 燈座 in Chinese anyway). Together with a 23W light bulb that Jin gave me last time I was able to take some better pictures of our beaded molecules. I have picked out some of them that looks pretty well...
I found that, in particular, some of the color combinations of beads don't look so well while others, like the ones shown above, seem much more aesthetically pleasing when taken into pictures. And, interestingly, even if all the settings remain, the color of the photoed object actually affects the color of background in the pics, for example the last photo has some kind of bluish background. I guess using background cloth with opposite (or conjugated?) color against the object may produce better photo. Think I'll get some other cloths with different colors for further experiments...