Conic sections in addition and multiplication graphs of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Compare two kinds of addition and multiplication graphs of the cyclic groups $$\mathbb{Z}$$ and $$\mathbb{Z}/n\mathbb{Z}$$:

1. group tables (with colored circles on a rectangular grid)

2. line graphs (with straight lines connecting points on a circle)

In any case one observes "implicit" lines, circles, ellipses and hyperbolas, especially as contour lines in the group tables and as envelopes in the line graphs.

Lines

• in addition group tables both for $$\mathbb{Z}$$ and $$\mathbb{Z}/n\mathbb{Z}$$, $$n=20$$:

Circles

• in multiplication group tables for $$\mathbb{Z}/n\mathbb{Z}$$, $$n=20$$: • in the addition line graphs for $$\mathbb{Z}/n\mathbb{Z}$$, $$n=20$$: Ellipses

• in the addition and multiplication line graphs for $$\mathbb{Z}$$ (for addition by $$20$$, $$50$$ and $$100$$ and multiplication by $$5$$, $$10$$ and $$50$$):  Interlude: Cardioids, nephroids, etc.

You may wish to compare these graphs to the apparent cardioids, nephroids, etc. in the multiplication line graphs for $$\mathbb{Z}/n\mathbb{Z}$$, $$n = 100$$, multiplication by $$2,3,4$$: Hyperbolas

• in the multiplication group tables for $$\mathbb{Z}$$ and $$\mathbb{Z}/n\mathbb{Z}$$, $$n=100$$: • surprisingly one can discover hyperbolas even in multiplication line graphs for $$\mathbb{Z}/n\mathbb{Z}$$, for example for multiplication by $$85$$ in $$\mathbb{Z}/256\mathbb{Z}$$: My questions are:

• By which general argument can be seen that one must not expect to "see" parabolas (the missing conic section) in any of these graphs?

• How to prove that the envelopes in the line graphs really are circles resp. ellipses?

• How to prove that the observed hyperbola-like contour lines in the multipication group tables really are hyperbola?