— for the impatient : a canonicalization algorithm —
All polygonal chains considered in this page are fixed-angle chains : their turn angles are fixed and can't be changed. The only way to move chains of this kind is via dihedral motion, that is motions where only the dihedral angles vary. A fixed-angle chain is said to be acute when \( \theta_i > 90^\circ \) for every \( i \), otherwise it is nonacute. A chain is monotone in some direction if every plane perpendicular to this direction cut the chain in at most one point.
A chain is said to be in a flat-state when all its segments are on a common plane. If it's possible to move the chain from any one of its flat-state to any other, using a self-avoiding dihedral motion, we say that the chain is flat-state connected. It is still unknown if every polygonal chain is flat-state connected. But it seems (see this paper and this video lecture) that the following cases are flat-state connected :
Also, this paper exposes a link between flat-state connected chains and chains that are in some way "producible" inside of a cone modeling a ribosome.
One common way to prove flat-state connectivity is to prove that there is an algorithm that can move the chain from any one of its flat-state to some canonical configuration (using a self-avoiding dihedral motion). If it's the case, then the chain is flat-state connected, because we can move it from the flat-state to the canonical configuration and then back to another flat-state by reversing the canonicalization movements. It is an example of an algorithm used to prove a theorem : if the algorithm works for every chain satisfying some conditions, then every one these chains is flat-state connected.
For the final page of this website, I wanted to implement something explaining one of the algorithms of the previously cited papers. What I did is this, it is an application which lets you draw a chain and then apply the canonicalization algorithm for nonacute chains described in section 3 of this paper. But unfortunately I lacked time, so this application is somewhat buggy and doesn't work every time, and I still have to investigate why. It is nonetheless helpful to understand the algorithm, I advise the reader to try it with this chain and see the result (I know it works for this one !).
