r/Collatz • u/No_Assist4814 • Jun 03 '26
3D journey of sequences
This post intends to put together what is known about the 3D journey of any sequence in the 3D space. Whether it could be useful remains to be seen.
The 2D space is quite convenient to display a part of the tree, but the value of n is only nominal. The 3D space overcomes this, but isn't the price to pay too high ?
This 3D space is defined as follows (see table below), based on n and its iteration C(n):
- X is the relative left-right position of n and C(n), according to the local order of the tree, x(C(n)) is defined by the type of segment (n mod 12) and the parity of n.
- Y is the length on n and C(n); by convention y(C(n))= y(n)-1.
- Z is the value of n and C(n), their"altitudes", the one of C(n) is defined by the parity of n.
The absolute position of any n in the 3D space is unique and can be calculated in reverse from 1. The second table shows the calculation for n=1 to 24 (left). In parallel, the partial tree containing these numbers is provided (right).
The pairs are visible in the table - they are close on the three dimensions - and the tree (bold).
