Strange connection between phi and pi

[Dr Finlay]

Me and a few friends at school the other day were wondering if pi and phi were somehow connected, After a while i managed to throw out

 

[math]\phi^{(\displaystyle\frac{\pi + \phi}{2})}= \pi[/math]

 

which manages to roughly approximate [math]\pi[/math]. I then found if you did

[math]\phi^{(\displaystyle\frac{\pi + \phi}{x})}= \pi[/math] with [math]x = 2.000811416[/math],

the equation exactly reached [math]\pi[/math]. But [math]x = 2.000811416[/math] seems too random to me, is there any connection between [math]\pi and \phi[/math] that would produce [math]x = 2.000811416[/math]?

 

On the slight chance you understood what i said, do you know where [math]x = 2.000811416[/math] can be derived from?

 

Cheers,

Rob

[Xyph]

Why not just simplify it further and say [math]x\phi = \pi[/math], so [math]x = \frac{\pi}{\phi}[/math]? I don't see what you're trying to do. What's special about [math]\phi^{(\frac{\pi + \phi}{x})}[/math] that you're picking it over a simpler operation?

 

There's always going to be some way to get from one number to another, so an equation that gets [math]\pi[/math] from [math]\phi[/math] with the aid of an arbitrary process and another irrational number isn't proof of anything.

[Dr Finlay]

I didnt think there was anything special about the equation. Just if there was anything special about the arbitrary number, so that the expression could be completely written in terms of phi and pi, rather than arbitrary numbers.

[Xyph]

x is always going to be defined by phi and pi, since they're both irrational numbers, so whatever you do you'll just end up saying something like...

 

[imath]\frac{\pi}{\phi} = x[/imath]

[math]x\phi = \pi[/math]

 

...which is obvious anyway.

[Dr Finlay]

hmm, i dont get [math]x = 2.000811416[/math] when i do pi/phi

[Anjruu]

He's not asking about the x value. He's wondering if there is a elegant and relativly simple way to relate pi and phi. Like a Euler's Identity for pi and phi, instead of pi, e, and i.

[Dr Finlay]

Could you rearrange

 

[math]

\phi^{(\displaystyle\frac{\pi + \phi}{2})}= \pi

[/math]

 

for [math]\phi[/math]?

 

Are there any other ways which relate phi and pi, my method is rather poor, and only approximates pi to a few decimal places.

[Xyph]

[math]2log_{\phi}\pi - \pi = \phi[/math]

 

hmm, i dont get x = 2.000811416 when i do pi/phi

Yeah, it's not the same x, I was just using it as an example of how x is always going to equal some function of pi and phi.

[Dr Finlay]

Ah right.

 

Cheers mate.

 

As i said, i didnt think there was anything special about the way my equation approximates pi, there are probably an infinite number of ways. I was just wondering if there was an elegant way to connect phi and pi. Maybe i should have just said that at the start:embarass:

[the tree]

Could you rearrange...

[math]

\pi^{\frac{-(\pi + \phi)}{2}}=\phi

[/math]

[Dr Finlay]

Cheers everyone;)

[Mobius]

There are plenty of relationships between pi and phi.

 

First of all if you have a circle of diameter phi then the circumference is pi*phi and is known as the golden circle.

 

Ancient Egyptian's wondered about the relationship between phi and pi, their ancient measurement the cubit was about pi/6 of a metre. they also found this was phi squared / 5.

 

to be fair, as mentioned, any number can arbitrarily be made into another number without any special connection. The thing is pi and phi do seem to have some unusual properties.

 

For example the Phibonacci brick (length and heigth phi units, width 1 unit).

Now if this brick is placed exactly inside a sphere the radius of that sphere is 1 and the surface are is 4*pi....

 

Just google all this stuff to find out more....

[Dr Finlay]

Thanks i'll have a look. I understand any number can be made into any other arbitrary number. I was just wondering if there was a way to derive pi from phi or phi from pi. I just thought there must be a way since these 2 numbers have unusual properties. All in all it was just a conversation at school so i wasnt expecting anything. Anyway it's filled in my "learn something new everyday" quota for the week.

[AL]

I'm sure you can have fun deriving simple relations between the two yourself. Pi is related to a circle (circumference/diameter). Phi is related to pentagons (diagonal/side) and pentagrams. So a good way to start is to inscribe a pentagon/pentagram in a circle and fool around with the trigonometry until you get a nifty relation, possibly involving sin and cos.

[Prince Blake]

There is an object called the Spiral of Life which has some very fascinating properties in relation to Pi and phi.  The object suggests that there exists some approximations to pi and phi which play a very important role in describing the object itself while not detracting in any way the significance of pi and phi themselves. 

A pi approximation using key measurements along the Lewin orbit 

The Lewin orbit is a location where the ever-expanding path and shifted orbital path conjoin with a high degree of synchronicity between them. The four vertices of the expanding path are intervals 77, 88, 99 and 112.   The vertices of the Lewin orbit in mid-shift are 77, 89, 101, and 113.   The key intervals forming an approximation of pi come from the intersection of expanding and orbiting paths..  An easy way to visualize these intervals is to imagine a bow and arrow at the vertices of the Lewin orbit (77, 89, 101, 113) with an additional interval - 88 - representing the grip location and interval 89 representing the arrow rest where the arrow is launched.  In this object 113 represents the serving where the arrow drawn in preparation for launch. 77 and 101 are where string is attached to bow at bottom and top respectively.   In this scenario, the sum of the four intervals along the bow  ( 77, 88, 89, and 101)  divided by the serving (113)  leaves us with 355/113 - a very close approximation to pi. 

Other Pi coincidences

The full shift of the Lewin orbit represents a shift in every vertex. In other words at mid-shift we have as vertices intervals 77, 89, 101 and 113 while full-shift is simply one more at each vertex resulting in 78, 90, 102 and 114.  The expanding intervals of the model are centered on intervals 1, 2, 3 and 4.  This simply suggests the first four intervals are tightly woven such that all are considered centered intervals of the object.  The product of the first interval of the Lewin orbit 77 and the centered core interval 4 and the fully-shifted interval on opposite side (north) 102 results in 31416.

In this model a new generation expansion is highly suggested beginning where the Lewin orbit comes full circle with the overlap of intervals 77 and 125.   Thus the first four intervals of the Spiral of Life correspond not only with the timing of intervals 125, 126, 127 and 128 but also 77, 78, 79 and 80 the sum of which is 314.   Dividing this result by the first Passover interval - 100 - and we reach the pi approximation 3.14 (Interval 100 is so-named because the new vertices of the orbital path "passes over" interval 100 and 101  before arriving at 102 to complete the full shift corresponding with the arrival of interval two.)

Finally a coincidence surrounding pi and phi

This is what prompted me reply to this forum. The location where a new generation expansion begins is where intervals 77 and 125 conjoin. Set as a decimal fraction that gives us .616.  No, it's not phi but it's very close.  But what's more, in exploring this fraction uniformly inward I happened along  a very close approximation to Pi which I hope you will find as interesting and eye-opening as I did.

The spiral's four arms expand according to the Golden Egg sequence.  It is a recursive sequence like the Fibonacci sequence only it begins with 0, 0 and requires adding one to each recursion.  This results in a largely prime column (save 9) whose intervals north are  3, 9, 13, 23, 37, 61...  and whose intervals south are (2+3), 5, 11, 17, 29, 47, 77...

If we examine the south column where 77 and 125 conjoin to form the Lewin orbit and explore the resulting fraction inward we are met with these results:

125 x .616  = 77

77 x .616 = 47.43

47.43 x .616 = 29.22

29.22 x .616 = 18.00

18.00 x .616 = 11.09

11.09 x .616 = 6,83

6.83 x .616 = 4.21

4.21 x .616 = 2,59

2.59 x .616 = 1.60

1.60 x .616 =.9833

.9833 x .616 = .6058

.6058 x .616 = .3731

.3731 x .616 = .2298

,2298 x .616 =.1415924131

A very nice approximation of Pi's decimal fraction!

 

[Rob McEachern]

φ=2cos(π/5)

[taeto]

If you also want to see an \( e \) you can have

\[ e^{\pi i/5} = \frac{\varphi + i\sqrt{3-\varphi } }{2}. \]

It is cheating, since both sides are just a \( \sqrt[5]{-1}. \)

Edited June 20, 2018 by taeto