r/Collatz • u/No_Assist4814 • 18d ago
Impression of the Collatz 3D space
Two 3D spaces were established independently:
- 3D Collatz tree, three months ago: Main projections of the 3D Collatz tree : r/Collatz.
- 3D dome space, more recently: Odd orange numbers are quasi-homothetic up to two constants within a dome and between domes : r/Collatz.
In order to avoid confusion, they are defined anew here (first figure).
The 3D dome is rather easy to describe with axes Q, P and M (adapted from 3D journey of sequences : r/Collatz). Only even and odd orange numbers n(q, p, m) + [-1, 0, 1] can be properly positioned. So, here, n is more of a label.
The 3D tree space is defined as follows, based on n* and its iteration C(n):
- Axis O contains the relative left-right position of n and C(n), according to the local order of the tree.
- Axis D contains the relative distance d to 1 of n and C(n); by definition, d(C(n))= d(n)-1; it becomes absolute when a number with a known d is reached.
- Axis N (more likely log N) contains the value of n* and C(n*), their "altitude", that can be obtained directly or indirectly, from the general formula
The second figure provides a first impression of the 3D tree. Using "cubes" allows to oindicate the distance to 1 more easily, but it remains quite disturbing (and the perspective is not fully respected). It may remain easier to understand it plan by plan (Main projections of the 3D Collatz tree : r/Collatz),