With the help of our friends over here:
https://mathematica.stackexchange.com/questions/6219/ulams-spiral-with-oppermans-diagonals-quarter-squares
We created Ulam's Spiral with Oppermann's diagonals
We noticed an East-West symmetry with the starting points of the SW and NE diagonals and a North-South symmetry with the starting points of the NW and SE diagonals. We drilled down to the center of the spiral and added 4 lines to show this relationship
Assuming each regular edge is 1 unit, we can calculate the distance each diagonal's starting point is from our new center vertex, add them together, and then calculate the average. We got a surprising result.
$\frac{1}{4} \left(\sqrt{\frac{5}{4}}+\sqrt{\frac{5}{4}}+\frac{1}{2}+\frac{1}{2}\right)=\frac{\phi }{2}$
where $\phi$ is the Golden Ratio
We know that there exists a Golden Spiral, but this result is unexpected.
Can anyone explain this?