Step 3 : Extend all possible ways

The title of this post refers to the tendency of algorithms in papers to have detailed explanation of every step except the most crucial one. Right now I am rediscovering this peculiar pleasure in structure generation.

As an example - or more as visual decoration - have this image:

which looks nice, but needs some explanation. The diagrams in boxes that look like parachutes (as one of my colleagues put it :) are simple representations of atoms connected by bonds. Each point is an atom, and a curved line connecting them is a bond.

These are grouped together by what structure they correspond to; which is shown on the right of each set of diagrams. What this shows, then, is the redundancy you get from a simple generator. If you connect all atoms like this:

  for atomA in atoms:
  for atomB in atoms greater than atomA:
    connect(atomA, atomB)

You quickly get a very large number of isomorphic structures. And then your process runs out of memory, in my experience.

Oh, and the numbers below each graph are the sum of the indegree/outdegree (the degree?) of the that vertex - or the number of bonds the atom is in, in other words. This seems similar to the ideas behind Morgan numbers, which I now understand a bit better.

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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.

The Gale-Ryser Theorem

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...

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 ...

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