Numbers In Nature: A Piece of Pi
By David Rusbarsky
rhubarb@csh.rit.edu
Throughout time numbers and their seemingly magical properties have fascinated
many people. Probably the most popular of these would be the number Pi. With its
seemingly endless trail of decimal points we can sometimes imagine that a pattern must
exist to explain the chaos. If there were a pattern, what would happen? What would that
mean? These questions can probably not be answered until a pattern is found and the
repercussions are already set in motion.
To help us understand pi, let us take a step back from the number and look at
where in the real world we find this number. We can find it all over in nature. In circles,
spheres, spirals, etc. What else can we find in nature? The Fibonacci numbers can be
found all over in nature as well. Could it be that these two numbers are somehow
related? Would it make sense to find out that there is a relationship between two
numbers that are commonly found in nature? Let us venture back to the number to take a
closer look.
Let us assume that the Fibonacci numbers will be grouped into sets of two. This
means that 1 and 1 will be a set, 2 and 3 will be a set, and so forth. If we were to look for
these sets in the digits of pi, spacing each of the numbers in the set evenly, we will
eventually find these the digits we are looking for, provided the digits of pi have even
distributions (ie, there is no pattern to the digits). This can be seen when we search for
the set 1,1 in Pi. We find the first 1 in the second digit of pi (ignoring the decimal). The
second 1 could then be found by jumping the same number of spaces (2).
314159…
Finding the second set (2,3), we can see that the jump size is 3:
31415926535897…
Finding the third set (5,8), we can see that the jump size is 1:
31415926535897…
The average jump should approach 100. This is because while we are searching
through all the possible jumps, there is a 1 in 10 chance of finding the number we are
looking for and a 1 in 10 chance of the number of spaces we have jumped so far will
result in the second number we are looking for. 1/10 * 1/10 = 1/100. As you can see, we
are off to a good start finding the first three sets in well under what the average should be.
This obviously shouldn’t mean much because we don’t know what the next jump will be,
but it makes our jobs of finding the sets a whole lot easier when they appear well below
this expected average jump size.
If we continue to go at this rate, we run into a problem when the Fibonacci
numbers start to increase past 8. Searching for single digit numbers in a single digit
number set is fine, but now we want to search for a multiple digit number in a single digit
number set. Using a technique I call folding (I believe the proper terms are Additive
Persistence [http://mathworld.wolfram.com/AdditivePersistence.html] and Digital
Root[http://mathworld.wolfram.com/DigitalRoot.html], although the process is slightly
modified to allow for 0s), we can condense any number into a single digit. Adding all the
digits of a number together to produce a new number does this. If that number is then a
multiple digit number, we can perform the process again and again until a single digit
results. One side note I should mention is that if a number is a multiple of 10 at the end
of any of the foldings, it is then converted to a zero so that zeros could be represented in
our number set. Since the next set we are looking for includes multiple digit numbers,
lets take a look at that as an example:
The next Fibonacci number is 5+8=13
Following our process, 13  1+3=4
4 is a single digit, so we can stop.
The next Fibonacci number is 8+13=21
Following our process, 21  2+1=3
3 is a single digit, so we can stop.
Therefore, the next set we are looking for is 4,3.
I will refer to the number of times you must do this process as the degree of
folding. Both of these numbers only required one run though the process, so they would
only have a folding degree of 1. There might be more rules to this process, but this gives
you the basic ides of how we will be searching for the Fibonacci numbers in Pi. The
other rules are still being looked at and being considered before I write about them. Oh,
and incase you were wondering, our new set can be found with a jump size of 8. In fact,
here is a look at the first 46 Fibonacci numbers in 23 sets including at what locations the
numbers are found in Pi. The Rn numbers are what I refer to as Rhubarb numbers and
they represent the jump size for that pair.
R0: 2
Fib pair (1,1) at 2,4
R1: 3
Fib pair (2,3) at 7,10
R2: 1
Fib pair (5,8) at 11,12
R3: 8
Fib pair (4,3) at 20,28
R4: 29
Fib pair (7,0) at 57,86
R5: 22
Fib pair (8,9) at 108,130
R6: 21
Fib pair (8,8) at 151,172
R7: 5
Fib pair (0,6) at 177,182
R8: 7
Fib pair (4,0) at 189,196
R9: 57
Fib pair (5,6) at 253,310
R10: 184
Fib pair (0,8) at 494,678
R11: 43
Fib pair (0,9) at 721,764
R12: 146
Fib pair (0,0) at 910,1056
R13: 55
Fib pair (2,3) at 1111,1166
R14: 4
Fib pair (5,0) at 1170,1174
R15: 1
Fib pair (4,0) at 1175,1176
R16: 582
Fib pair (7,0) at 1758,2340
R17: 49
Fib pair (8,9) at 2389,2438
R18: 111
Fib pair (8,8) at 2549,2660
R19: 2
Fib pair (7,6) at 2662,2664
R20: 1
Fib pair (4,0) at 2665,2666
R21: 9
Fib pair (5,6) at 2675,2684
R22: 98
Fib pair (0,8) at 2782,2880
I found it interesting that the average of these numbers is actually close to 63 and
not 100 and only 4 of Rhubarb numbers were over 100, but as I said before, this is
inconsequential unless we know all the sets, which is impossible due to the idea that the
digits of pi do not end.
If we could find a pattern or equation to the Rhubarb numbers, would this then
mean there is a pattern to pi? Another interesting development of these numbers is that
the Lucas numbers seem to start appearing in a slightly different pattern. The Lucas
numbers are derived in the same manner as the Fibonacci numbers, but the first two
numbers are 2 and 1 producing the following numbers:
2,1,3,4,7,11,18,…
2 is located at location 1 of our number set, or by jumping 1. The next Lucas
number can be found by jumping 2, then 4, then 8 (double the jump for each number)
using our previous trick of finding multiple digit numbers in a single digit set.
2,3,1,8,29,22,21,5,7,57,184,43,146,55,4,1,582,49,111,2,1,9,98,…
The digits of pi are actually determined by a formula. This formula, however, has
changed through the years to get more precise digits. Could the Rhubarb numbers also be
expressed with an equation that might change over time to get more precise digits?
When the equation for Pi changes, would the equation for the Rhubarb numbers not
change as well? Could any of these Rhubarb number be incorrect due to coincidence?
Maybe one of those 1 in 100 chances of finding the two numbers we were looking for
occurred and we were knocked off the correct path when we were supposed to look a
little farther down the chain of Pi digits.
Another interesting note to make is the appearance of Pascal’s Triangle in the
Fibonacci number sequence. These are two number sets that are commonly found in
nature and in the world around us, and they are related. Are we surprised by this? Why
should we be surprised to find the Fibonacci numbers in pi then?
What does all this mean? Are we stuck in some sort of mad loop looking for
patterns in patterns until we come full circle (if we ever come full circle), or are we
finding some sort of intricate connections between numbers that are commonly found in
nature through the use of mathematics? I would like to think that the latter is true.