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.


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

enter image description hereenter image description here


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

enter image description here

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

enter image description here


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

enter image description here

enter image description here

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 $2,3,4$:

enter image description here


  • in the multiplication group tables for $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$, $n=100$:

enter image description here

  • surprisingly one can discover hyperbolas even in multiplication line graphs for $\mathbb{Z}/n\mathbb{Z}$, for example for multiplication by $85$ in $\mathbb{Z}/256\mathbb{Z}$:

enter image description here

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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.