Skip to main content

Cuneane Maps

So, to continue about combinatorial maps, here are some more intricate diagrams. Firstly, an embedding of cuneane with a 4-cycle as the outer face:


The permutations below are just for reference. Anyway, with this embedding, the cycles of ϕ are (0,8,23,21,5)(1,2,11,7)(3,4,17,15)(6,12,9)(10,14,18,22,13)(16,20,19) which does indeed have one for each face, including the boundary. If you use a different embedding, naturally you get a different map:


Which is quite different, and has ϕ of (0,6,10,3)(1,4,20,22,9)(2,14,16,5)(7,8,13)(11,12,23,19,15)(17,18,21) - again, 6 cycles for the 6 rings. And one to rule them all, and in the darkness bind them, of course. The cycles of darts are shown in this composite image:


Some of the triangular faces are missing the dart labels, as they were getting too crowded. Well, the whole thing is too crowded, but still. Highlighted in red in each are the darts corresponding to the outer face of the other embedding. Not sure what it means, though.

Finally, some references:
1) doi:10.1.1.149.3832 citeseer link - it's a dissertation, not a paper, but interesting.
2) Signatures of combinatorial maps (direct link to pdf) S Gosselin, G Damiand, and Christine Solnon. Damiand is the author of some of the images on wikipedia about combinatorial maps...

Comments

Post a Comment

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...
2 comments
Read more

chalky

I wanted to show something that hints at the things that the new architecture can afford us: This is using a Java2D graphics Paint object to make it look like chalk...kindof. It's a very simplistic way of doing it by making a small image with a random number of white, gray, lightgray, and black pixels. edit: it doesn't look so good at small scales some tweaking of stroke widths and so on is essential.
2 comments
Read more

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 ...
2 comments
Read more