Phi
Besides p and e, you may be interested in the "golden rectangle". This is a ratio called Phi ~1.618 that ancient peoples discovered as the key to art and beauty, and is applied to the study of beauty in the human face, architecture and art, etc. It is repeatedly found in nature, because it arises naturally from the way things grow. The exact value is irrational, ie the decimal digits go on for ever, but it is one of the easiest constants in nature to calculate/estimate.It is defined exactly as
Phi=1+1/Phi.
Put another way,
1 / 1.618034.. = 0.618034..
The number Phi is equal to it's own reciprocal plus one.
So: if x=1+1/x, then x2-x-1=0
Then using the quadratic formula Phi= ( 1 ± sqrt(5) ) / 2.
The two solutions are:
1.618033988749894848204586834365638... and
-0.618033988749894848204586834365638...
suggesting the duality of relationships.You can find this value to the accuracy of your calculator by repeating the following sequence: +1, 1/x,
repeated 15 or 20 times, or until the values quit changing.
You can save time by pre-setting the total to a close value, but even if you start with some other number, the result will quickly zero in to our magic number. If your calculator doesn't have a 1/x key, often the two keys divide= do the same thing, and you may have to hit = after + 1, so it becomes +1=/=,+1=/=,+1=/=...Other Ratios
Consider x=N+1/x, found by repeating +N,1/x. The next after Phi would be x=2+1/x which is 1± sqrt(2) then.... = (N± sqrt(N2+4))/2Fibonacci Numbers
One place you will find the magic ratio Phi is Fibonacci numbers. This is the series of numbers, start with zero and one, where the next number is the sum of the last two. As these number get larger, the ratio of one to the next closes in on our magic ratio 1.618034... You can see these number using a calculator with a memory function and the sequence:Start by entering 1,
then repeat this sequence:
(M+), (+)(MR)(=)
keys 1 M+ +MR= M+ +MR= M+ +MR= M+ +MR= M+ +MR= display 1 1 2 2 5 5 13 13 34 34 89 memory 0 1 1 3 3 8 8 21 21 55 55 Lucas Numbers
A similar series is the Lucas numbers, also found in nature. Starting with the 5th Lucas Number (=11), they are very close to the powers of our magic number 1.618.... You can calculate Lucas numbers the same way, but start with 2 in memory, instead of zero.
keys 2 M+ 1 M+ +MR= M+ +MR= M+ +MR= M+ +MR= display 2 2 1 1 4 4 11 11 29 29 76 memory 0 2 2 3 3 7 7 18 18 47 47 The 100th Lucas number is 792070839848372253127 and the 100th root of this is
1.61803398874989484820458683436564, the reciprocal of
0.618033988749894848204586834365638On an 8 digit calculator, if you enter the 16th Lucas number 2207, and hit the square root key 4 times, the result is 1.618034 . Because log(Phi) is about 1/5, the Nth Lucas number will be about N/5 digits long, and, it seems, equal to Phi^N with about 2N/5 digits accuracy.
Both Lucas and Fibonacci numbers are symmetric. They can be worked backwards:
Note the sign of every other number is negative in the negative part of the series.
Fibonacci 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 Lucas -76 47 -29 18 -11 7 -4 3 -1 2 1 3 4 7 11 18 29 47 76 Links
Fibonacci Numbers
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