Understanding a group of transformations of the plane $\mathbb{Z} \times \mathbb{Z}$

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?

• I'm not sure what you want to ask. Perhaps to present such transformations into groups are relators? Aug 16, 2018 at 14:17
• What's a "relator"? Do relators compare to group actions like groupoids to groups? Aug 16, 2018 at 14:57
• Aug 16, 2018 at 15:27
• Does every subset of $\mathbb{Z} \times \mathbb{Z}$ qualify as a spiral? Given a spiral, what is its "rectangular eye"? Aug 17, 2018 at 15:49
• I tried to make this clear above. So the answer is no: Every pair in $\mathbb{Z} \times \mathbb{Z}$ qualifies as a spiral. Aug 17, 2018 at 16:00

I don't particularly understand what you are asking for. When you write $T^k_l(S^i_j) = S^{i+k}_{j+l}$ Are $S^i_j$ and $S^{i+k}_{j+l}$ two particular spirals (if so, which ones?) or is it the same spiral sampled differently?

At any rate, here are a few thoughts.

Visualizing an arithmetic sequence in this way is , as you say, not a new idea. The Ulam Spiral visualizing the primes is well known.

You seem to only be looking at sequences of integers $at^2+bt+c.$ These may visually seem to produce multi armed spirals. This depends on good rational approximations to square roots. At larger scale another structure corresponding to a better convergent may pop out visually.

Below is your representation of the non-negative integers and the same thing rotated by a quarter turn. The green lines indicate the squares. The third picture is the second spiral along with the smooth spiral given by the polar equation $r=\frac{-\theta}{\pi}$ with the points indicating that the integer $t$ is represented by $[\sqrt{t}\cos(-\pi \sqrt{t}),\sqrt{t}\sin(-\pi \sqrt{t})].$

Now the squares appear exactly at the integer points of the $x$-axis. You may check that the points corresponding to the sequence $t^2+t$ are slightly displaced from the $y$-axis. With this transformation it makes sense to consider sequences such as $t^2+t+1/4.$ These , of course, appear exactly on the $y$-axis at the points between integers.

I'll claim without justification that the visual appearance of a quadratic sequence is about the same for the rectilinear spiral and the smoother one I am using. The rotation rate is essentially the same and the sampling is fairly sparse.

In your model and mine the triangular numbers $\frac{t^2+t}{2}$ seem to fall into a $3$ armed spiral. I'll explain that briefly by instead looking at the sequence $\frac{t^2}{2}$ which is visually very similar.

The angles are $\frac{t}{\sqrt{2}}\pi$ and taking every third one provides a rotation of $\frac3{\sqrt{2}}\pi \sim 2.1213\pi.$ Since $.1213 \pi$ is about $\frac{\pi}{8}$ we are not surprised that there are about $16$ points in a rotation of one arm. However with more points it turns out that $17$ arms is even nicer. See that at the end.

The pattern for $2t^2$ is less obvious. The points seem pretty scattered. In fact (just as with others) there is no one right pattern,Since $7\sqrt{2} = 9.8994\dots$ and $17\sqrt{2} =24.041\dots$ we see that $7$ somewhat jerky arms have about $10$ points per turn and $17$ fairly smooth arms have about $24$ points per turn. $41$ arms would be even smoother and $99$ (given enough sample points) would be especially smooth since $99\sqrt{2}=140.00714\dots.$

Finally, here is the obviously $3$ armed spiral for $\frac{t^2}{2}$ expanded for more points. Suddenly $17$ arms seems more obvious

• Thanks a lot (especially for the nice pictures). Your answer doesn't quite answer my question (obviously the question has not been clear enough), but accidentally (?) another question of mine at MSE: math.stackexchange.com/questions/2884854/… Aug 17, 2018 at 9:08
• Yes, $S^i_j$ and $S^{i+k}_{j+l}$ are supposed to be two different spirals (which ones is explained in my question). Aug 17, 2018 at 9:11
• A question into the blue: 17 is an astonishing number, because the 17-gon was the first one surprisingly be proved to be constructible. Is there a deeper connection between these two occurrences of the number 17? Aug 20, 2018 at 12:37
• I don't think so 2/3,5/7,12/17,29/41,70/99 are increasingly good approximations to the square root of 1/2. I can't say that I see a seven arm spiral but maybe if it was drawn out? Aug 20, 2018 at 17:33
• Where do these numbers 2/3,5/7,12/17,29/41,70/99 come from? I'm stuck. (Or are you just kidding me? You're welcome - my question is probably nonsense.) Aug 20, 2018 at 17:41