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Let me share a kind of visualization of group tables which is especially well suited for cyclic groups like $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$. In these groups you can easily give colors to each group member $k$ with

  • hue = red if $k > 0$ (for $\mathbb{Z}$) or $k \leq n/2$ (for $\mathbb{Z}/n\mathbb{Z}$)

  • hue = blue otherwise

  • lightness = goes from 0 to 1 for $|k|\rightarrow \infty$ (for $\mathbb{Z}$) or for $k \rightarrow n/2$ (for $\mathbb{Z}/n\mathbb{Z}$)

So $k=0$ always looks black while $k = \infty$ resp. $k=n/2$ looks white.


My question is:

Which specific number- or group- or ring-theoretic facts can be learned from looking at these group tables?


It comes as no surprise that all addition group tables for $\mathbb{Z}/n\mathbb{Z}$ look essentially the same, independent of the nature of $n$ (be it prime or whatsoever):

enter image description here

But also the multiplication group tables exhibit quite regular patterns. But this time there are at least two of them: those with a distinguished black/white pattern at the center and those with a distinguished red/blue pattern:

enter image description here

A closer look at consecutive $n$ reveals another detail: the black/white and the red/blue patterns at the center flip every second step:

enter image description here

So far, so good. What remains interesting is the "limit pattern" which is approached for $n \rightarrow \infty$. It looks something like this (for $n=64,128,256, 251$):

enter image description here

As you may guess there is not a unique limit pattern, but only a kind of pattern whose details depend on $n$ (mainly its size), especially the distinctness and size of "subcenters" (next to the "main center" at $(\frac{n}{2},\frac{n}{2})$), e.g. of the four subcenters in the cases above which you may immediately see (near the centers of the four quadrants).

Note again that the existence and visibility of such subcenters are independent of the nature of $n$ – just compare $n=256$ (a power of 2) and $n = 251$ (a prime number).

Here is the multiplication group table for $n=512$:

enter image description here


For the sake of comparison here are the addition and multiplication group tables for $\mathbb{Z}$ itself (for $|k| < K$, $K = 10,100$:

enter image description here

enter image description here


To repeat my question:

Which specific number- or group- or ring-theoretic facts can be learned from looking at these group tables?

To ask one specific question:

How can the hyperbolas be explained which can be observed at the center of each plot, be it for $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$?


For another sake of comparion: See here some group tables with red = blue:

enter image description here

enter image description here

enter image description here

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  • $\begingroup$ I definitely remember another question with the same image of the multiplication table for 512 (or maybe 1024). I'll see if I can find it... $\endgroup$ – Mark S. Feb 19 at 16:53
  • $\begingroup$ @MarkS.: Hope you'll find it soon. (I will not be astonished by the image per se, because these images are very straight-forward. But the context will be interesting.) $\endgroup$ – Hans-Peter Stricker Feb 19 at 16:57
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    $\begingroup$ It was in this answer to a different question . It doesn't really answer the question you posed here, though. $\endgroup$ – Mark S. Feb 19 at 17:00
  • $\begingroup$ What a beautiful link! So many cute examples. $\endgroup$ – Hans-Peter Stricker Feb 19 at 17:02
  • $\begingroup$ The multiplication table image looks almost exactly like what you get if you plot the complex entries of the discrete Fourier transform matrix. (en.wikipedia.org/wiki/DFT_matrix) This might be a hint about the continuous limit. $\endgroup$ – Display Name Feb 20 at 20:53
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It turns out that the patterns can be seen more clearly when choosing another color scheme: for $k = \lfloor n/4 \rfloor$ choose red, for $k = \lfloor 3n/4 \rfloor$ choose blue, for the other values a shade of gray indicating the distance to $n/2$. For $n=128$:

enter image description here

One observes that the big square is divided successively into squares of sidelength $n/k$.

Furthermore one may observe that for example the red dots at the top left really lie on a hyperbola. They are placed at the grid cells $(32,1), (16,2), (8,4), (4,8), (2,16), (1,32)$ thus fullfilling $j = 32/i$.

For $n=257$, i.e. a prime number, the grid patterns vanishes, but the hyperbolic structure remains intact:

enter image description here

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Concerning the last, i.e. the second question which concerns ((sub)centers of) hyperbolas:

For $\mathbb{Z}/n\mathbb{Z}$ the visible centers of hyperbolas are placed at points $ \frac{n}{k}(i,j)$ for $k < \log_2 n$ and $0 \leq i,j \leq k$.

(Note the modular/toroidal structure of the graph.)

The "size" of the hyperbola shrinks with $1/k$, its "distinctness" is maximal along the diagonals.

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An even better way to plot the group table of $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$ – and better comparable to the $\mathbb{Z}$ case – is by putting $(0,0)$ to the center (instead of to the top left). This is how the tables look for $n=32,64,128$ and for $\mathbb{Z}$:

enter image description here

Another "advantage" is, that one always has a "black cross" through the origin $(0,0)$ – independently of $n$, especially of the primeness of $n$. E.g. for $n=127$ which is a prime number (compared to $n = 128 = 2^7$):

enter image description here

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