Computational Combinatorics

Month: July, 2014

Boron and Buckyballs

Recent news from the world of chemistry: the result is really the experimental observation of a special new molecule. People call it a “Boron Buckyball”, but this irritates me since we know that the buckyball is a specific fullerene on 60 vertices. In particular, it is the smallest fullerene that satisfies the independent pentagon rule. This new Boron molecule is called “Borospherene” and is quite interesting in its own right, as seen below:

The borospherene molecule. Credit: Zhai et al.

The borospherene molecule. Credit: Zhai et al.

The structure is very interesting when you consider it as a spherically-embedded (i.e. planar) graph: there are many triangular faces which create a cube-like structure, two of the faces of this cube are 6-faces, and the other four are 7-faces! These interesting heptagonal structures are particularly interesting. In the figure above, it appears to be a unit-distance embedding, and this creates a rotation in the opposing 6-faces.

Let’s dig more into the structure of this object, but also into the computational part of the experiment.
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On Reproducibility

Recently, Stephen Hartke spoke about our work on uniquely K_r-saturated graphs, and afterwards he spoke to a very famous mathematician, who said, roughly:

“Why did you talk about the computation? You should just talk about the results and the proofs and hide the fact that you used computation.”

While this cannot be an exact quote, I believe it is a common attitude among mathematicians. While the focus is on solving problems, the use of computation is seen as a negative, so the final research papers make little mention of their computational methods. This is a huge problem in my opinion. While computations can report exact results, and sometimes even prove results, every execution is an experiment. It is unknown before the execution what the results will be, or how long it will take.

Computational combinatorics is a combination of mathematics, engineering, and science: We prove things, we build things, and we experiment. Since computer proofs are experiments, it is important that they be reproducible. Today, I want to discuss a bit about how we can improve our presentation of algorithms and computation in order to make results more reproducible.
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