Introduction

MathOverflow is a Q&A website, which serves as an online community of mathematicians. While browsing through MathOverflow posts, I stumbled upon a rather peculiar result discovered by Simon Plouffe between 1974 and 1979. I'll borrow the statement verbatim.

Let the multiplication graph \( \frac{n}{m} \) be the graph with \( m \) points distributed evenly on a circle and a line between two points \( a, b \) when \( a \cdot n \equiv b \mod m \). These graphs look random but by carefully choosing \( n \) and \( m \) one finds intricate patterns.

So what are those intricate patterns? Well, defying all expectation, it turns out that these graphs trace out cardioids!

Multiplication Graph

Once you get all the graphical boilerplate taken care of, we can draw up a Modular Arithmetic Graph using the following pseudo-code!

                # These uniquely define the graph
                N = 40
                M = 139

                # Draw congruent lines
                for start in range(M):
                    for stop in range(M):
                        congruent = (start * N) % M == (stop) % M
                        if congruent: draw(start, stop)
                        else: continue
            

Running through that workflow, I managed to recreate some Modular Multiplication Graphs featuring cardioids!

Resources and References

If you're interested in the math behind these Modular Multiplication Graphs, I recommend checking out a fun Mathologer video on the topic. From there, you can work your way up to Simon Plouffe's original paper along with his other works.