I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\mathbb{N}$ on $\mathbb{Z}\times \mathbb{Z}$ in different ways and highlighting the numbers that have a given property (belong to a given set, are in a given sequence).

You can find a little tool with which I'm playing around here.

What I came up with has nothing to do with the initial approach (i.e. with arithmetic sets) but with transformations of $\mathbb{Z}\times \mathbb{Z}$. It's possibly interesting, but I'm missing conceptual tools to understand and explain it.

What I observed was that different spirals (i.e. regular bijective mappings $m: \mathbb{N} \rightarrow \mathbb{Z}\times \mathbb{Z}$ with $m(0) = (0,0)$) seem to be related by some kind of "quasi-rotations", combined with some scaling, translation and "disturbance" in the middle of the spiral. These transformations can be watched "in action" and indeed "feel" like rotations, sometimes accompanied by some characteristic intermediate "shrinking" (have a look).

I'd like to put these findings into mathematical terms and understand how they relate to each other and to the parameters of the spirals.

Let a spiral $S^i_j$ be parametrized by the width and height of its rectangular "eye", i.e. pairs of integers $(i,j) \in \mathbb{Z}\times \mathbb{Z}$. **The "eye" of a spiral is a rectangle of integer side length with one corner in the origin, around which the spiral winds clockwise.** This is the eye of the spiral $S^5_2$:

Each transition $T^k_l$ between spirals can be described by a pair of numbers $(k,l) \in \mathbb{Z}\times \mathbb{Z}$ such that

$$T^k_l(S^i_j) = S^{i+k}_{j+l}$$

Obviously, the transitions $T^k_l$ form a group. A transition seems to be related to

an angle of rotation $\frac{n\ \pi}{2}$ (see $S^i_j \leftrightarrow S^{i\pm 1}_j$, $S^i_j \leftrightarrow S^i_{j\pm 1}$) (depending on what and how?)

the relative size of a "disturbed" zone in the middle of the affected spiral (depending on what and how?)

some scaling and translation of "patterns" outside the disturbed zone that remain "intact" otherwise (depending on what and how?)

possibly an isolated translation of the upper-right half plane (see $T^{-2}_1$ which sends $S^3_0$ to $S^1_1$)

*[What's possibly not characteristic for a given transformation – but possibly depends on the spiral being transformed – is the intermediate "shrinking" factor which becomes visible only when watching the transformation "in action" (see $T^0_1$ sending $S^1_0$ to $S^1_1$, or $T^2_0$ sending $S^1_1$ to $S^3_1$, which both are not accompanied by a rotation).*

*Compare this scenario (as a vague analogy) with the physical scenario of "Big Collapse followed by Big Bang"]*

My question is:

How can I treat these ephemeral quasi-rotations in the context of "hard" group theory? Where in group theory is a place for transformations of the form "rotation + X" with a hard to grasp X?

pairin $\mathbb{Z} \times \mathbb{Z}$ qualifies as a spiral. $\endgroup$