Tuesday, March 5, 2013

Short summary of zome-type superbuckyball part IV: Tori and miscellaneous

Lastly, we will address some other examples of this method of constructing graphitic structures from C60s. We will start with planar tori. All of the zometool models of these tori reported here are composed of entirely blue struts.

Threefold torus⊗C60 with g=1: C216 (C3v)



Fourfold torus⊗C60 with g=0: C224 (C2h)



As noted parenthetically, the rotational symmetry of this structure is actually twofold only.

Fivefold torus⊗C60 with g=0: C280 (C5v)



Sixfold torus⊗C60 with g=0: C336 (D3v)



Hyper cube⊗C60 with g=(0,3,1): C1536 (Th)



Perhaps this hypercube should be also classified as polyhedron in the previous post, since it IS a regular polyhedron in 4D. As can be seen, there are three different kinds of struts (two blues with different lengths and one yellow) in this structure. It is thus more difficult to find a reasonable set of g parameters that suits all of their intricate geometric relations. The CNTs in the outer layer are bent to accommodate this incommensurability. I have to point out that, in graph theory, 4D hypercube cannot be represented by a planar graph. This fact leads to considerable difficulty, if not impossible, in constructing corresponding graphitic structures with out previous approach of using the inner part of TCNTs. However, with the zome-type construction scheme this is nothing different than other graphically simpler structures.

Dodecahedron with V-shape edges⊗C80 with g=1: C4960 (Ih)



Note that one has to at least use C80 (or larger Ih-symmetric fullerenes) instead of C60 for the nodes, since the yellow struts joining at the same node are nearest neighbors to each other. Total fifty C80s were used, twenty for the (inner) dodecahedron and thirty for the (outer) edges. If I make it to the Bridges this year in Enschede, I'll bring a beaded molecule of this model with me.

Last but not least, how can I not play with trefoil knot?

Trefoil knot⊗C60 with g=(0,1,3): C912 (D3)



Unfortunately, so far I have not thought of any general scheme to construct arbitrary torus knots, as trefoil knot is only the simplest nontrivial case of them. In principle as long as the structure (or the space curve as for knots) can be constructed with zometool, there is also graphitic analogs of it and presumably beaded molecules as well. This concludes this series of posts. I'm currently working on a beaded molecule of C60⊗C60 with g=1 posted previously. I'm about half way there and I might talk about some specific beading strategies of this kind of structures later on.

2 comments:

Bih-Yaw Jin said...

I like the symbol, C60⊗C60. It can be viewed as a kind of convolution.

How about the fractal structures like C60⊗C60⊗C60 or even, C60⊗C60...⊗C60? I mentioned this in the short description for the exhibition in JMM held in San Diego early this year.

Bih-Yaw Jin said...
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