# Graphical multiplication tables for $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}$

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 $$p= 11,29,97$$:

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

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":

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 $$n \rightarrow \infty$$:

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?)

• 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? – Tobias Kildetoft Nov 8 '18 at 8:57
• Which ones exactly do you mean? – Hans-Peter Stricker Nov 8 '18 at 9:00
• For $p=11$ the ones labelled as $\times 3$ and as $\times 4$ are identical. – Tobias Kildetoft Nov 8 '18 at 9:02
• Actually, they are identical in pairs, so something has gone wrong. – Tobias Kildetoft Nov 8 '18 at 9:08
• 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. – Hans-Peter Stricker Nov 8 '18 at 9:15