Is the formula for Phi unique or should we say, “Hey, it’s just an expression!”
It’s been noted by some who say they can “demystify phi” that phi is just one of an infinite series of numbers that can be constructed from the following expression using the square root (√) of integer numbers:
(1+√n) / 2
It just so happens that you get phi when you let n equal 5. Let n be other integers and you get a series of numbers whose squares (see Phi2 in table in green) each exceed their root by a difference (see Δ in table in blue) that increases by 0.25 for each number in the series, as shown below.
Phi, being the 5th one in the series, just happens to be the one that produces a difference of 1 with its square, leading to the unique property that it shares with no other number:
Phi + 1 = Phi 2
| 1 | n | 2 | x = (1+√n)/2 | x2 | Δ | 1/x |
| 1 | 1 | 2 | 1 | 1 | 0.00 | 1 |
| 1 | 2 | 2 | 1.207106781 | 1.457106781 | 0.25 | 0.828427125 |
| 1 | 3 | 2 | 1.366025404 | 1.866025404 | 0.50 | 0.732050808 |
| 1 | 4 | 2 | 1.5 | 2.25 | 0.75 | 0.666666667 |
| 1 | 5 | 2 | 1.618033989 | 2.618033989 | 1.00 | 0.618033989 |
| 1 | 6 | 2 | 1.724744871 | 2.974744871 | 1.25 | 0.579795897 |
| 1 | 7 | 2 | 1.822875656 | 3.322875656 | 1.50 | 0.548583770 |
| 1 | 8 | 2 | 1.914213562 | 3.664213562 | 1.75 | 0.522407750 |
| 1 | 9 | 2 | 2 | 4 | 2.00 | 0.5 |
So does this demystify phi, making it just one of a series of phi-like numbers? Not necessarily, as this is only one aspect of phi’s unique properties. Phi is also the only number that produces a difference of 1 with its reciprocal:
Phi – 1 = 1 / Phi
This is the key to its relationship to the golden section, which is based on sectioning a line in a way that fulfills two requirements:
A = B + C
and
A/B = B/C
A is to B as B is to C, where
A is 161.8% of B and B is 161.8% of C, and
B is 61.8% of A and C is 61.8% of B
Let n be any integer other than 5 and you won’t find the same pattern of consistent differences as shown above or the unique reciprocal and additive properties of phi.
Insights on phi’s formula in the table above contributed by Joseph Conklin.
also if you take the squareroot(1+squareroot(1+squareroot(1+squareroot(1+squareroot(1+…))))) you get phi.
? Square root of what?
It’s an infinite series. He’s saying that you keep on taking the square root and adding one each time.
Phi is the second in an infinite sequence of n-nacci constants which all satisfy the equation;- F + 1/F^n =2
In the case of n=2 we get;-
Phi + 1/Phi^2 =2 ;(Where Phi = The Fibonacci constant)
When n=3 we get the Tribonacci constant
n=4 gives the Tetranacci constant
n=5 gives the Pentanacci constant
and so on ‘ad infinitum’… producing an infinite sequence of constants that converges on the value 2.
Phi is certainly unique in that it is the only n-nacci constant the produces a difference of 1 with its reciprocal, but it is still just the second in a well documented
sequence of n-nacci constants which of each have unique properties.
Regards,
Ted.
Ok so where do all these other n-nacci constraints reproduce in nature?
I am like God But I am not Dog….
For all intiger n, phi^n plus phi^(n plus 1) equals phi^(n plus 2)
yolo
???
I found another property of phi.
Σ π( 1/2 – 2/3 + 3/5 – 5/7 + 7/11 – 11/13 … a/b ) = Φ
such that a is the last denominator, b is the next numerator, and both a and b are consecutive prime numbers.
Is there a name for this property? I can’t find a reference for it.
Very cool
If you put every other odd number into the equation for n you get the next integer from that last. When N=1,5,9,13,17 ect.. Then the result ^2 – the result = 0,1,2,3,4.
To me this sounds like where the (0,1,2,3,4) ‘demarcations’ are, that those are the Whole Integer Harmonic Steps, with each step being a resonant harmonic of the n-Series given.
And I think this is why Phi (the penta-series, one could term it) is unique, in that at its harmonic periods, it manifests or shows within it ANOTHER Integer, the 1(One), and then the 0,618 is like the ratio equivalence between the harmonic steps. To me it seems like its….. i guess you could say “Fractally Harmonic”? Idk if thats the best way to put it in words, but it makes sense in my head, lol.
Anyway, just found this site….. def liking it!!! I’ve been learning and researching everything and anything I could about esoterica, Golden Section geometry, Sacred Geometry, and the like, for probably a little over 20 years now. And let me tell you…. what i Overstand today, is MUCH different, or rather, has more depth and meaning associated with the “maths” and geometries/shape/topologies, than the Understanding i THOUGHT i had back then after just a few years of high school finding out about this kinda stuff and looking into it.
And life experiences definitely have helped to shape that change over the years. Its more Real to me now, the dynamics and interplays of the numbers and the shapes and structural geometries….. i mean, its really the ONLY thing ‘out/in there/here’ in existence!!!
I think the Integers were created/are created, by the One Divine Overmind/Godhead’s Desire to Create for the sake of experience, and that they are almost a by-product of that “initial/inertial” process, but also the rules or guide-bars by which it supports itself, and by extension, Every Thing and non-Thing.
For anyone interested in deep Spiritual but also Geometric/Mathematical and beautiful philosophy and art, I highly suggest a few people to look into if you havent heard of them:
WALTER & LAO RUSSELL – A husband/wife couple. Lao that started the University of Science & Philosophy originally as the Walter Russell Foundation. Predicted the existence of at LEAST 5 different elements before they were “discovered” or isolated and known to science and academia. Friends and contemporary with Nikola Tesla and ‘Mark Twain’ or Samuel Langhorne Clemens; one of the most AMAZING and brilliant inventors in known human history, and one of the most well-known and eccentric authors, respectively. Walter Russell was a Polymath, meaning basically an expert in many fields. And he was almost completely self-taught, though not without the help of friends and correspondence with people in science, art, philosophy, religion, and more, all over the world. He was a SUPERB sculptor, with many of his pieces shown on the http://www.philosophy.org site, the University of Science & Philosophy’s official website. He was also told by Nikola Tesla, after showing him his book The Universal One, and the principles and scientific data inside, that he should lock up his material for 1000 years, and have it not open until much time had passed because he believed that the world and Humanity would not be ready for it, or be able to use it responsibly yet, if they could fully interpret it at all yet (which we have since found out, since he DIDNT lock it up, and thought that this information should be freely given to Mankind that we may have the information to unimprison ourselves in the illusions we hold to be Truth. And MUCH research has since proved MUCH of the Russell’s information either ON POINT correct, or at least on the right track in some other areas.
JAMES “BUDDY” DOUGHERTY – created The Dougherty Set. A logarithmic geometric set of drawings or connections or structures, that show the finer structures we have discovered in nature, and the rules behind its creation. He is also into Sonobiology, the study of biological effects/growth/decay solely by different specific sound frequencies (which are just are terms for an invisible 3D structured energetic field pertubations), very close to Cymatics. Also interested and speaks about Tao and Eastern thought and philosophy as well.
He is also a ‘student’ of the Russell’s work (among others).
VIKTOR SCHAUBERGER – Austrian forester, inventor, natural philosopher (seeing a pattern here… hmm, maybe thats the point 😉 ) and he discovered a PLETHORA of secrets about the way water works, moves, and actually is truly Alive. Or, at least, it can be. It an also be diseased and sick and dead. And he creates electric generators and motors and levitational/inertial-driven devices/crafts. He also helped revolutionize the foresting industry with his principles and technology.
How about this:
(((phi)^1/phi)^1/phi)……^1/phi) / phi = phi – 1
there are infinitely many (^1/phi) in the blanks above. Looking good yeah?
Interesting. The entire term (((phi)^1/phi)^1/phi)……^1/phi) converges on 1 though. So what it says in simpler terms is 1/phi = phi-1, the basic expression of phi’s unique reciprocal and additive properties.
I don’t see why you would have to say Phi^1 instead of just Phi. Any number to the 1st is itself.
The formula is meant to be read as (phi)^(1/phi), not ((phi)^1)/phi.
Pi equals 5cos^-1(phi/2)
Why we don’t use simple formula?
1/x=x-1
It give exact Phi…
Phi = .5 + √1.25
“Phi is also the only number that produces a difference of 1 with its reciprocal:
Phi – 1 = 1 / Phi”
Not so, 1-Phi also works, which follows from the quadratic:
x-1 = 1/x
x^2 – x = 1
x^2 – x -1 = 0
->quadratic formula time<-
(1±sqrt(1-4*1*(-1)))/(2*1)=x
x={1.618…, -0.618…}
I propose this -0.618… should be known as Phil, the Phi-Like Number.
Alternatively, just change the sentence to say "Phi is also the only POSITIVE number that produces a difference of 1 with its reciprocal" (emphasis added).
True. This and other Phi facts are covered on our math page at https://www.goldennumber.net/math/.
As Evan says:
1/Phi = Phi – 1
1 = Phi^2 – Phi
a b c
0 = (1)Phi^2 + (-1)Phi + (-1), where a – 1, b = -1 & c = -1 in Phi = (-b ± (b^2 – 4ac)^(1/2))/2*a
b b a c a
Phi = (-(-1) ± ((-1)^2 – (4*(1)*(-1))^(1/2))))/2*(1)
Phi = (1 ± (5)^(1/2))/2 = 1.6180339887498948482045868343656… & – 0.6180339887498948482045868343656…
Phi Phi
In the above, the site ignored my spaces that would have located the letters a b 7 c over the correct ( ). Ooopa 7 = &
Hello can someone give me phi formulas in other bases, like 9,7 0r 5 ? It would be very interesting to compare them in different bases, I think !
Also you get a bunch Of phi formulas by typping « phi formula » in the free Wolfram Alpha app such as : 2 cos(pi/5) etc…