2
$\begingroup$

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 $\mathbb{Z}/n\mathbb{Z}$, $n=20$:

enter image description hereenter image description here


Circles

  • 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


Ellipses

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

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


Hyperbolas

  • 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?

$\endgroup$

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.