Structure: Minor Hexagons

Cell types are interesting, but they simply reflect a modulo 6 view of numbers.  More interesting are the six internal hexagons within the Prime Hexagon.  Like the Prime Hexagon, they are newly discovered.   The minor hexagons form solely from the order, and type, of primes along the number line.  Perhaps they have no significance … however, as they do form I suspect they must connect to deeper things in mathematics.   I’ll offer evidence below supporting this notion, but for now I’ll simply describe them.

Prime Hexagon:  Minor hexagons and their polarities
Prime Hexagon: Minor hexagons and their polarities

Numbers fill the hexagon by spiraling until a prime number is met, then it jumps to the next minor hexagon and begins to spiral again.  Below are the spins for the first 80 numbers; 2 (0 and 1) spiral blue, meet prime 5, spiral purple briefly, meet 7 and spiral red. The bands are pretty short because primes are common in the early numbers, but the grow rare as numbers grow large and the bands broaden.

Numbers and their 'spin' states in the Prime Hexagon.
Numbers and their ‘spin’ states in the Prime Hexagon.

Now consider your favorite number – 14,999,999,969.  What color does it spin?  You might guess any one of the colors above because the spin is not calculable (though you can find the value with my algorithm) … it’s essentially random with 1:6 odds of being any given color.  Of course, it’s not truly random since primes do not appear randomly, but in human practice and understanding, they do appear randomly.

So now I will attempt to show the minor hexagons are significant.  This is not easy as they are linked to the nature of prime numbers, and nothing is easy about the nature of prime numbers.  But I begin with this assumption: if the hexagons participate in the Universe in any way other than haphazardly, they must be demonstrably congruent to something organized.  That is, if I can show they are organized (not random) in relation to  some other thing, then primes and the thing are linked.

Here is an example.  Say you own a casino and you’ve been told that someone tampered with one of your die (dice).  So you take all the dice and toss them.  They all appear to come up randomly, except for one.  This die rolls 6’s each time, 10 rolls in a row.  Has it been tampered with?  Maybe.  Then one of your employees say’s “Hey, that’s the die Cheatin’ Charlie was using!”   A true scientist, you ignore the statement that the most notorious dice cheater of all time was using it and you simply look at the facts.   However, the statement is enough to make you redouble your efforts.   You give it another 10 throws and still it comes up 6.  You will never know for certain the dice is fixed, even after 50 throws, but the certainty goes up with each toss, as is your certainty Cheatin’ Charlie had been at it again.

Similarly, I have a six colored dice in the form of the hexagon.  If I take a known, logical sequence of numbers, say 10, 100, 1000, 10000, and look at their spins in the hexagon, the resulting colors associated with each number should appear random – unless the sequence I’m investigating is linked to the nature of the prime numbers.  

That is, if the powers of 10 all returned with blue spin, or as a series of rainbows, or evenly alternating colors or other non-random results, then I’d say prime numbers appear to have a linkage to 10.  I may not know what the the linkage is, just that it appears to exist.

Back at the casino, I might not know how Cheatin’ Charlie fixed the dice, but with each throw I’d grow more confident he did fix the dice.

I have catalogued the spins for all numbers below 2.6*10^16*, so I can assign color spin values to each member of a sequence and visually inspect for suspicious behavior.  This is not as simple as it sounds.   Below I’ve taken random dice rolls and assigned them a corresponding spin color.  You will likely notice random looks clumpy, which is a little puzzling.  (Weird but true… in fact, I made a bunch of 6 colored dice to fully convince myself that random looks this way.)

Random dice returns with Prime Hexagon spin states associated with them to allow one to 'see' what random looks like.
Random dice returns with Prime Hexagon spin states associated with them to allow one to ‘see’ what random looks like.

To screen sequences for further study, I assigned members of dozens of sequences their spin states and visually inspected their relative colors to see if I can literally ‘see’ patterns. Unfortunately, some sequences grow quickly, so are poorly suited for finding any trends because they quickly exceed my ability to assign spins. Others grow slowly, so few primes separate members and can’t randomize the appearance of spin colors.

Other sequences contain numbers with fractional values, like pi or e.  Like a number line, I assume non-whole numbers fall between integers. As for rounding up or down or truncating, see Phi, not Pi, and Why I Truncate to Determine Integer Values.

I’ve only screened for very obvious deviation from random. Other, subtle indications of non-random behavior may well reside in the color bars I’ve assembled; I did not look for any such behavior.  To be clear, by obvious I mean straightforward and simple.  And not not associations found by sophisticated statistical analysis.  By obvious, I do not mean my findings are obviously correct or obviously astounding.

To date, I have found only one number sequence that visibly produces non-random results: pi and its powers, shown as truncated for display purposes.  I will post the root high precision results, along with raw data, in the near future.

The spin states for the powers of pi.  The Prime Hexagon is an integer environment, so pi powers are truncated.
The spin states for the powers of pi. The Prime Hexagon is an integer environment, so pi powers are truncated.

If you are like me, you might think this look perfectly random — except mathematical randomness is clumpy as we saw above. Dice roll two in a row.  Dice roll three in a row. Adjacent powers of pi, say 31 and 97, do NOT land in the same hexagon.  I can’t say they won’t ever do so, but I’ve rolled the pi dice 33 times and it has not done so.  The odds of this happening randomly are less than 1/4 of one percent.**

I believe these data suggest prime numbers are linked in some way to pi.

 

* Technically, while my database goes to 2.6*10^16, I’ve only saved one value every 10 billion numbers because to store every value would require  colossal amounts of disk space.  I use these values as seeds to regenerate numbers should I need them.

**My computers are calculating toward roll #34 and I should have those results in mid-March.  To get to #35 would take 1 more year given my current computer cluster.  I should also caveat rolls #32 and #33 — I used many threads to get their values and integrated the results by hand, twice.  I will not yet stake my life on their accuracy, though I believe them to be so.  I will remove this caveat in the near future.  (Update: On March 6, my single thread calculations, which do not rely on my hand integration of results, have now confirmed roll #32).