Hmm. Not my favourite solution, but I think that the isomorphism checks can be done in batches, resulting in actual isomorph spaces, like this (for C5H10):
Well, I had been hoping to reject isomorphs without doing an all-v-all check with the UIT. Sadly this didn't work - at least with the tests that I thought of.
So, what I did was all-v-all check the structures produced from a single fragment combination (a single partition) as I assumed that there would only be duplicates within the children of a partition, and not between partition descendants.
Um, well the partitions are just lists of numbers. There may be all sorts of interesting patterns in them, but mostly of interest to number theorists, not chemists :)
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.
This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG). So, GR says that given two partitions of an integer ( p and q) there exists a (0, 1) matrix A iff p* dominates q such that the row sum vector r(A) = p and the column sum vector c(A) = q . As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example: The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q , the former dominates. One possible matrix is at the top left with the row sum vector to the right, and th...
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 ...
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So, is this the result of a generation for C5H10 or has it already been filtered?
So, what I did was all-v-all check the structures produced from a single fragment combination (a single partition) as I assumed that there would only be duplicates within the children of a partition, and not between partition descendants.
Hmmm. I'll make a diagram..
Maybe there are recognizable patterns?
Eg:
10 = [[4, 4, 1, 1], [4, 3, 2, 1], [4, 2, 2, 2], [3, 3, 3, 1], [3, 3, 2, 2]]
In fact, it is also [[7, 1, 1, 1], [6, 2, 1, 1], [5, 3, 1, 1], [5, 2, 2, 1]] but those are rightly rejected for having valences greater than 4.