>Going further, I note that the ideal bond length was given to me with just two significant digits, suggesting that an error of one part in a billion is much smaller than our uncertainty in measuring the “true” bond length in the first place.

Bonds are flexible and can hold mechanical stress, so there is only a "preferred" length, not a guaranteed length.

One of my proudest contributions: https://www.ncbi.nlm.nih.gov/pubmed/24265211

Protein design and fold prediction was being held back by using fixed (equilibrium) bond angles in proteins, and if you have enough CPU time, you can optimize those values (they often end up centering around non-equilibrium angles for the particular atom types) to get better predictions.

Do you happen to know if this is related to what was done with AlphaFold? It sounded like they moved from fixed values to a probabilistic model but I'm not sure if bond angles was the property they were measuring.

Great question. I don't know.

To put numbers on this, a typical approximation for bond energy as a function of stretch would be E = 500x^2, where x is the stretch in Angstroms and E is the energy change in units of thermal noise at room temperature (about 0.6 kcal/mol). A bond length deviation of 0.01 Angstroms is already negligible, because its associated stretch energy is far below thermal fluctuation levels. Parts per billion are totally invisible.

This touches on something I find fascinating: that in the pure world of perfect maths there's a lot of near misses, things that nearly work out but don't quite, but that if we have a world where a bit of error is allowed (our bendy molecular world, with our limited consciousness) then an enormous host of possibilities opens up. My personal theory is that this is exactly why we find ourselves in a reality that is slightly fuzzy with an awareness that's alright but not perfect.

For anyone interested in polyhedra I'd like to highlight the wonderful antiprism project:

http://www.antiprism.com/

And the web-based Polyhedronisme for a quick fix: http://levskaya.github.io/polyhedronisme/

Finally (personal plug) I've been scratching my own polyhedral itch with a project that aims to cover similar ground but in a more interactive, visual way:

https://github.com/IxxyXR/Polyhydra

(Example output: https://photos.google.com/album/AF1QipM029IxQ5P5FacaGywBPlxS... )

One factor that unifies all three projects is conway operations discovered John "Game of Life" Conway: https://en.wikipedia.org/wiki/Conway_polyhedron_notation later investigated and expanded upon by George Hart - who is mentioned in the opening of the main article.

The linked nature article.

https://sci-hub.tw/10.1038/s41586-019-1185-4

Actual atoms in molecules also vibrate all the time, so the provided bond length he talked about should be seen as an equilibrium bond length.

A graphic illustration of the molecule is here https://www.nature.com/articles/s41586-019-1185-4/figures/8

Reminds me of the time I made a very regular looking polyhedral model that wasnt and had the hardest time tracking it down.