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

Question

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.

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Lines

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

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Circles

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

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

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Ellipses

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

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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 $2,3,4$:

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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 $\mathbb{Z}/256\mathbb{Z}$:

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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?