Perhaps my favorite number is phi. So when I began to explore the hexagon, I eagerly set out to see if one of phi’s semi-magical properties would make sense in the context of the hexagon, or conversely, if some mystery of the hexagon would become clear looking at phi.
I started with phi’s cousin, the Fibonacci Series, which has nice integer values that land squarely, if unremarkably, in the hexagon. Phi, an irrational number, would be more complicated to fit into the integer defined cells of the hexagon. Should I round up to get the integer? Round down?
Recalling the Fibonacci Series has a pisano period of 24; I wondered if that property might hold for the incremental powers of phi as well. If it had a pisano period also, the modular for its members should have the same value. If I rounded to the nearest, and the pisano period had errors, then I would know I should truncate. That is what I found. Phi and its members have a pisano period if the resulting fractional numbers are truncated. If one instead rounds to the nearest number, the modular values for the first two terms are flipped compared to the 24th and 25th values as seen below and look more like Lukas numbers.
For this reason I chose to see numbers in the hexagon as quantum, and truncate off the decimal values to determine which integer cell they land in.
A couple interesting asides about Phi. Powers of phi grow increasingly closer to whole numbers. Even powers create a number just a tiny bit shy of a whole number. Odd powers create a number a smidge above a whole number. So close, in fact, one must use high precision tools to be certain one has the right integer. Phi really is cool.
Also, when the colors of the hexagon cells (not spin color) are charted for each of the phi powers, wave-like patterns appear. These can be split, and more wave patterns appear, see below. Very pretty.
As for phi’s spin values, it was a bit of a disappointment. If there is some pattern there, its invisible to the screening I’m currently doing.