The Magic Ratio


Explored

.61803

The ancient greeks somewhere knew about this certain ratio which had interesting properties. They defined it by taking a line segment, and dividing it into two parts, in such a way that the ratio of the larger portion to the original segment is the same as the ratio of the smaller segment to the larger. This ratio was used in many places in the ancient world, from DaVinci's art to the Great Pyramids.

Fibonacci Sequence: 1,2,3,5,8,13,21,34,55,89,144.....
The Fibonacci sequence is the set of numbers aquired when one takes first the numbers 1 and 2 and adds them to get 3, and from that point on adds the the last two numbers of the series to get the next number in the series.

The directly relates to the magic ratio: the ratio of two adjacent numbers in the series approaches the magic ratio, which is an irrational number (this means that it, like pi, continues forever on the right side of the decimal point).

1 / 2 .5 2 / 1 2
2 / 3 .66_ 3 / 2 1.5
3 / 5 .6 5 / 3 1.66_
5 / 8 .625 8 / 5 1.6
8 / 13 .6153846 13 / 8 1.625
13 / 21 .6190746 21 / 13 1.1653846
21 / 34 .617647 34 / 21 1.6190746
34 / 55 .61818__ 55 / 34 1.6176471
55 / 89 .6179775 89 / 55 1.61818__
89 / 144 .6180555 144 / 89 1.6179775

The table above represents the ratio of adjacent numbers in the fibonacci sequence, and demonstrates quick convergence on the Magic Ratio. The left hand side of the table represents the fibonacci sequence read left to right (Dexter), which the right hand side of the table represents the fibonacci sequence read right to left (Sinister).

Here I would like to introduce the abbreviation Mr for Magic Ratio, and Dex for Dexter an Sin for Sinister. SinMr is Sinister Magic Ratio, and DexMr is Dexter Magic Ratio. Mr alone is Dexter. TrueMr refers to the inexpressible, irrational number that we would get if we were able to continue this table forever.

Note that both the DexMr and SinMr are converging on a TrueMr 1 apart! Now, notice the portion of the ratio to the right of the decimal point. Note the diagonal symmetry!

Notes: the _ is used to represent the traditional bar over the top of the numbers, to indicate repeating digits to infinity, i.e. 1.6666666 is 1.6_ and 1.61818181818 is 1.618__. There is also imperfect diagonal symmetry, but I believe this is due to imprecision in the calculator used to perform these functions.

The convergence toward TrueMr above could be expressed as: 

Fibonacci[n] / Fibonacci[n+1]
for DexMr, and: 
Fibonacci[n] / Fibonacci[n-1]
for the SinMr, where n is a position in the Fibonacci sequence.

DexMr² + DexMr =~1 (approaches 1). I am comfortable in saying, then, that: TrueMr² + TrueMr = 1.

A Step Farther Out

What about two numbers seperated by one in the Fibonacci sequence? What properties do they have?

For lack of a better term, I will temporarily designate this "Wierd Ratio", and abbreviate it Wr.

DexWr = Fibonacci[n] / Fibonacci[n+2]

SinWr = Fibonacci[n] / Fibonacci[n-2]
1 / 3 .3_ 3 / 1 3
2 / 5 .4 5 / 2 2.5
3 / 8 .375 8 / 3 2.6_
5 / 13 .3846153 13 / 5 2.6
8 / 21 .3809523 21 / 8 2.625
13 / 34 .3823529 34 / 13 2.6153846
21 / 55 .381__ 55 / 21 2.6190746
34 / 89 .3820224 89 / 34 6176471
55 / 144 .3819444 144 / 55 2.618__

Here, as in the Magic Ratio, both SinWr and DexWr are converging to TrueWr. However, unlike Mr, these are not one apart. But, compare SinWr and SinMr - and notice that they are one apart!

SqrRt(DexWr) + DexWr =~1 (The Square Root of DexWr plus DexWr converge on one). Very peculiar!

This is the end of my research. I am not a mathematician, so I don't know what I've discovered, or rediscovered, and what is currently known. I intend to continue, and will present what I find.

---Doctor Who