I recently found out that Mathematica can plot polar coordinates, and I really liked polar plotting. Today, I found something surprising. The $n$ th prime gap is defined as $$g(n)=p_{n+1}-p_n$$ I made a polar plot of $(g(n),n)$ and this is the result:
This surprised me, because prime gaps tend to be in some diagonals more than others, and there are some areas left untouched. Here is a plot with from $-10000$ to $10000$:
And this pattern continues to hold. My question is:
Why do prime gaps tend to be in some "lines" in the plot more than others? What is special in them?
Another thing:
Does this tell us anything about prime gaps?
First, note that there are $22$ spokes since $\pi \approx 22/7$. If you were to zoom out (much) farther, you'd see another pattern emerge with $355$ spokes.
With that established, now observe that every increase by $2$ in the prime gap progresses $7$ spokes counterclockwise, since $$2 \cdot \frac{180}\pi \approx 2 \cdot \frac{180}{\frac{22}7} = 7 \cdot \frac{360}{22}.$$
If you were to lay out your data in a more standard histogram, you would see that there are many small gaps and fewer larger gaps. See this question, for example: Prime gaps distribution.