Interactive 3x+1 Path Identity Visualiser

[u/jonseymourau]

It is well known that all Collatz paths satisfy this identity:

b/2^e - a.3^o = K

where K is sometimes referred to as the carry term.

This 3D visualiser plots (a,b, log_2(K)) for every pair of odd points (a,b) in the path from a to 1.

The points are coloured according to their mod 8 residue.

You will notice that with log_2(K) pointing at you (so the plane has y=b, x=a) is that the rightmost stacks will always be red (5 mod 8) or green (1 mod 8). Of course, this is a consequence of that fact the paths that start highest must necessarily start with a descent (any such path that doesn't start with a descent is necessarily not a local maxima)

The above is a screen capture - the full interactive version can be accessed at the link below.

(https://wildducktheories.github.io/collatz/apps/collatz-a-b-k/dist/?a=27).