Skip to main content

Blowing Carbon Bubbles : Expanding 2D Fullerene Layouts to 3D

The concentric face layout code is working well enough now to handle the larger fullerenes - such as that old favourite, C60. Since coloring the vertices by equivalence class is not always terribly informative, here is a view of the ring equivalence classes :

Where C60 is on the left, and a more colourful C70-D5h is on the right. One difficulty, however, is to understand the symmetries of these structures when they are distorted like this. The further away from the center of the layout, the more stretched the rings become.

So, an obvious next step was to 'blow up' these 2D layouts into 3D. It turns out that is possible, with a combination of inverse stereographic projection and Jmol's minimize command. The first step is necessary since minimizing the 2D coordinates (with a z-coord of zero) just shrinks the diagram down in the plane. Here are before and after shots of these steps:

Clearly the inverse-projection does not give very good 3D positions for the atoms, but they are on the surface of a sphere, and the bonds don't cross. The minimization gives a much better looking version, although there are still dents and a lot of asymmetry.

Next step is to write out the equivalence class colours (vertex/face?)  into a Jmol script, so that they can be visualised in 3D. Oh, and one last thing - it was necessary to scale the flat diagram down so that the radius was closer to a unit circle. If it was contained inside the circle, then the expansion was only a hemisphere. Also, the points were expanded from the centre outwards by a small factor, to improve the bond distances.

Comments

Popular posts from this blog

Adamantane, Diamantane, Twistane

After cubane, the thought occurred to look at other regular hydrocarbons. If only there was some sort of classification of chemicals that I could use look up similar structures. Oh wate, there is . Anyway, adamantane is not as regular as cubane, but it is highly symmetrical, looking like three cyclohexanes fused together. The vertices fall into two different types when colored by signature: The carbons with three carbon neighbours (degree-3, in the simple graph) have signature (a) and the degree-2 carbons have signature (b). Atoms of one type are only connected to atoms of another - the graph is bipartite . Adamantane connects together to form diamondoids (or, rather, this class have adamantane as a repeating subunit). One such is diamantane , which is no longer bipartite when colored by signature: It has three classes of vertex in the simple graph (a and b), as the set with degree-3 has been split in two. The tree for signature (c) is not shown. The graph is still bipartite accordin

Király's Method for Generating All Graphs from a Degree Sequence

After posting about the Hakimi-Havel  theorem, I received a nice email suggesting various relevant papers. One of these was by Zoltán Király  called " Recognizing Graphic Degree Sequences and Generating All Realizations ". I have now implemented a sketch of the main idea of the paper, which seems to work reasonably well, so I thought I would describe it. See the paper for details, of course. One focus of Király's method is to generate graphs efficiently , by which I mean that it has polynomial delay. In turn, an algorithm with 'polynomial delay' takes a polynomial amount of time between outputs (and to produce the first output). So - roughly - it doesn't take 1s to produce the first graph, 10s for the second, 2s for the third, 300s for the fourth, and so on. Central to the method is the tree that is traversed during the search for graphs that satisfy the input degree sequence. It's a little tricky to draw, but looks something like this: At the top

1,2-dichlorocyclopropane and a spiran

As I am reading a book called "Symmetry in Chemistry" (H. H. Jaffé and M. Orchin) I thought I would try out a couple of examples that they use. One is 1,2-dichlorocylopropane : which is, apparently, dissymmetric because it has a symmetry element (a C2 axis) but is optically active. Incidentally, wedges can look horrible in small structures - this is why: The box around the hydrogen is shaded in grey, to show the effect of overlap. A possible fix might be to shorten the wedge, but sadly this would require working out the bounds of the text when calculating the wedge, which has to be done at render time. Oh well. Another interesting example is this 'spiran', which I can't find on ChEBI or ChemSpider: Image again courtesy of JChempaint . I guess the problem marker (the red line) on the N suggests that it is not a real compound? In any case, some simple code to determine potential chiral centres (using signatures) finds 2 in the cyclopropane structure, and 4 in the