The group of benzene, against how much error is forgiven

One of the figures on point groups from coordinates: A molecule turned in space with its group recovered from the turned coordinates, and what the group settles on its own.

Three essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The tolerance is a decision

Four distortions of benzene, none larger than six hundredths of an ångström, against nine tolerances. Each cell is the symbol the search reports. The top row is the exact structure, which is D6h at every tolerance from a thousandth to a third of an ångström — so everything below belongs to the distortions rather than to the search. The cell picked out in warning colour is a step that is not a subgroup step at all.

The same molecule against a wider range of tolerances than the grid above uses, from a thousandth of an ångström to three tenths. At the tight end the search returns the group the coordinates exactly have; at the loose end it returns a group the molecule does not have at all. Nothing about the structure changes across that row — only the amount of error being forgiven.

The count of candidate operations that pass, at each tolerance, for the four distortions. Every curve rises and none falls, which is guaranteed by the form of the test rather than observed. This is the monotone quantity; the symbol assigned to what was found is not one.

The group of a molecule that will not hold still

The point group recovered from a deliberately distorted molecule at a sequence of tolerances. The group grows as the tolerance is loosened, exactly as the feasible group grows as the temperature is raised, and neither sequence has a step that is the right one independently of what is being asked.

How much symmetry is left

The result that made this necessary. Four distortions of benzene at nine tolerances, with the group the search reports in each cell. Every column is the same four structures; every row is the same structure. Nothing about the molecules changes across a row, and the name does.