Inspired by Burkard Polster's beautiful video on Times Tables, Mandelbrot and the Heart of Mathematics I wondered how this graphical approach to visualize the multiplicative structure of finite rings could be extended to infinite rings, especially to $\mathbb{Z}$.

Here are a few examples of graphical times (= multiplication) tables for $p= 11,29,97$:

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Enlarged for the first numbers for $p=97$:

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Note that one immediately sees that $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime has no zero divisors: no lines end at $0$, the point at the bottom.

My simple idea was: Just project $\mathbb{Z}$ on the unit circle and just draw the "times tables":

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Note that and how and why the diversity of forms for finite rings gets lost!

Note further that the $\times +n$ and $\times -n$ tables tend to look the same for $n \rightarrow \infty$:

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I wonder:

  1. Has this graphical approach been taken already?

  2. What might its didactical/pedagogical/educational values be?

  3. Mathematically: How do the graphs for finite rings – among each other and with the graph for the infinite ring $\mathbb{Z}$ – relate? (In which terms would one define such relations, how would one analyse them?)

  • $\begingroup$ The tables on the top for multiplication by $3$ and $4$ are identical. Have the elements been rearranged in some systematic way between the diagrams? $\endgroup$ – Tobias Kildetoft Nov 8 '18 at 8:57
  • $\begingroup$ Which ones exactly do you mean? $\endgroup$ – Hans-Peter Stricker Nov 8 '18 at 9:00
  • $\begingroup$ For $p=11$ the ones labelled as $\times 3$ and as $\times 4$ are identical. $\endgroup$ – Tobias Kildetoft Nov 8 '18 at 9:02
  • $\begingroup$ Actually, they are identical in pairs, so something has gone wrong. $\endgroup$ – Tobias Kildetoft Nov 8 '18 at 9:08
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    $\begingroup$ Not really: $3 \times 4 \mod 11 = 1$ so there is a line from $3$ to $1$ in the $\times 4$ table. But there is also a line from $1$ to $3$ in the $\times 3$ table - obviously. The problem so is that the direction of the lines is not indicated: with arrow heads, the two tables would not be identical. $\endgroup$ – Hans-Peter Stricker Nov 8 '18 at 9:15

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