Recurrence grids

First you have a set of points that make lines like so:

This represents the line from (x1, y1) to (x2, y2) and we give it an initial value:

Then you have a recurrence relationship over the previous 4 variables like this:

P1 under the recurrence goes to:

And then:

And the pattern repeats making the regular octagon with inradius [1,0]...

Of course there are infinitely many possibilities for the recurrence grid formulas I just thought this was a nice one...

**For general n-gons**

The general recurrence extending the one above for octagons with theta =pi/4 to a general case is:

So a pentagon would use a theta of 2*pi / 5, etc... Constants multiplying the bottom 2 entries of the recurrence grid make various sorts of spirals

Just a random example:

The first 7 lines generated:

I'm not sure what to make of this pattern of lines but I think it shows the patterns can be complex...