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 so-called intricate patterns? Well, defying all expectation, it turns out that these graphs trace out cardiods!

Creating a Multiplication Graph

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

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

# Let's start off with some lables
labels = range(M)

# Add lines
for num1 in labels:
    for num2 in labels:
        if (num1 * N) % M == (num2) % M: draw(sym1, sym2)
        else: continue
            

Running through that workflow, I managed to re-create some Modular Multiplication Graphs featuring cardiods!

Resources and References

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

Comment Section