Which rings close on a double bond and which do not

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Two 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 double bond a ring cannot hold

The residual against ring size, for cis and trans, on a logarithmic scale. The rings that close and the rings that do not are separated by more than two orders of magnitude, so the threshold does not depend on where the line is drawn. At the ordinary angles a cis double bond closes a ring from seven carbons up and a flat trans one needs nine.

The largest torsion each ring will close on: 180° is a flat trans arrangement and 90° is a π bond turned off outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction.

The twist the ring forces on the bond, ring size by ring size. A torsion is the one internal coordinate that can change a molecule’s symmetry without changing any bond length or angle, and here it is the coordinate the closure spends: a smaller ring can only close by twisting the double bond further out of plane.

A ceiling that rises where the measurements fall

Another ring size that cannot do what larger and smaller ones can. A closure constraint that switches on at a particular ring size, computed rather than remembered, is the shape of most of what this field has to say.