"Purity Spiral" is a neologism for a process that's quite visible in online spaces but was less obvious in the past.
I suppose the name comes from "Death Spiral" or "Flat Spin" or somewhere like that.
Discussion forums, especially political ones. are quite susceptible to polarization between two "sides".
At first you have L/C/R, with Moderates on the L and R sides (C for center). But let's go with X/Y now, because otherwise we could get distracted by US stuff.
So it's (X, XM, C, YM, Y).
You can apply the "Overton Window" model naturally at this stage.
Two things happen as they start to polarize:
1. X and Y separates a bit, so X moves towards a little towards "Extreme X and XM moves towards where X was before. Ditto for YM and Y.
2. The discussions get nasty.
Sometimes (and we'll go with this because it simplifies things) as X moves towards EX, and Y moves towards EY, one side will shrink and then depart due to the nastiness.
Meanwhile, C shrinks to effectively nothing, , because Center/Moderate people don't change much, and on any topic whatsoever, they leave as the extremists come in.
At this point the Overton Window is separated into two (or three if C stays).
(also take a look at "Horseshoe Theory" (that one has a wikipedia page))
Here's the odd part that leads to the most obvious "Purity Spiral" cases:
After most of the old Center and one side departs (let's say only X is left), the forum will have descended into "groupthink", and will keep moving towards its now singular extreme position. So the "C" has become as hated as "Y" was (this is a sure sign it's now out of control) and what's left is more or less (EX, X, XM).
Now the Purity Spiral kicks in.
As a natural side-effect of Groupthink, the online space often keeps moving towards its extreme position.
Our "X" group starts testing each other to make sure they hold the (ever shifting) orthodox position, and will mercilessly attack previous friends if they don't agree with the current extreme position in every detail.
So soon it's (EX, X). At some point it's barely practical to move further into EX (maybe one more step into EEX), but it's not unlikely that "X" itself (the original core) is destroyed.
(Note: I just discovered there's now a wikipedia page on this - probably better to read that than this post /lol)
Anyway, given time people get attacked for quite trivial things.
And while it's not obvious from the way I set this up, the topics can be quite trivial. Here's a copy/paste from the wikipedia article (in wikipedia it's a link to an external article:
Gavin Haynes: "How knitters got knotted in a purity spiral". /lol.
BTW this might be an example where they started with only one side. You don't need more than one side to get a Purity Spiral, but my personal experience is limited to 2-sided cases.