Phi (Φ) and pi (Π) and Fibonacci numbers can be related in several ways:

## The Pi-Phi Product and its derivation through limits

The product of phi and pi,

1.618033988… X 3.141592654…, or 5.083203692,

is found in golden geometries:

Golden Circle | Golden Ellipse |

Circumference = p | Area = p |

Ed Oberg and Jay A. Johnson have developed a unique expression for the pi-phi product (pΦ) as a function of the number 2 and an expression they call “The Biwabik Sum,”a function of phi, the set of all odd numbers and the set of all Fibonacci numbers, as follows:

p Phi = 2² {1 | + [ (2/3) / (F1+F2 Phi) + (1/5) / (F3+F4 Phi) – (1/7) / (F5+F6 Phi) ] |

– [ (2/9) / (F7+F8 Phi) + (1/11) / (F9+F10 Phi) – (1/13) / (F11+F12 Phi) ] | |

+ [ (2/15) / (F13+F14 Phi) + (1/17) / (F15+F16 Phi) – (1/19) / (F17+F18 Phi) ] | |

– … } | |

= 5.083203692…. |

This relationship was derived after Oberg noticed an interesting relationship between pi and phi while contemplating geometric questions related to the location of the King and Queen’s burial chambers in the Great Pyramid, Cheops, of Giza, Egypt, the design of which is based on phi.You can access the complete paper published by Ed Oberg and Jay A. Johnson, The Pi-Phi Product, in Word, or the Pi-Phi Product in Excel to see their formulation illustrated numerically.

## Trigonometric functions relating phi (Φ) and pi (Π)

Divide a 360° circle into 5 sections of 72° each and you get the five points of a pentagon, whose dimensions are all based on phi relationships.

Accordingly, it shouldn’t be too surprising that phi, pi and 5 (a Fibonacci number) can be related through trigonometry:

Or, a much simpler way involving, contributed by Dale Lohr:

Pi = 5 arccos (.5 Phi)

Note: The angle of .5 Phi is 36 degrees, of which there are 10 in a circle or 5 of in pi radians.

Note: Above formulas expressed in radians, not degrees

Alex Williams, MD, points out that you can use the Phi and Fives relationship to express pi as follows:

5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi

Robert Everest discovered that you can express Phi as a function of Pi and the numbers 1, 2, 3 and 5 of the Fibonacci series:

Phi = 1 – 2 cos ( 3 Pi / 5)

## Pi and Phi in the Great Pyramid of Egypt

Another interesting relationship between Pi and Phi is related to the geometry of the Great Pyramid of Giza. This relationship connects dimensions of the Great Pyramid to both Pi and Phi, but it is not known with certainty whether this was an intentional aspect of its design, whether its design was based on Pi or Phi but not both, or whether it is a simple coincidence. It relates to the fact that 4 divided by square root of phi is almost exactly equal to Pi:

The square root of Phi (1.6180339887…) = 1.2720196495…

4 divided by 1.2720196495… = 3.14460551103…

Pi = 3.14159265359…

The difference of these two numbers is less than a 10th of a percent.

See the Phi, Pi and the Great Pyramid page for more details.

## Pi squared (Π²) and 987

Pi squared (Π²) is 9.8696…, which, if you round to 9.87 and ignore the decimals, is 987, the 17th number of the Fibonacci series. (Contributed by William Erman.)

## More on the relationship of Phi squared and Pi

If you’re looking for other interesting ways to relate pi and phi, 6/5 * Phi^2 = 3.1416, which approximates pi. (Contributed by Steve Lautizar.)

j says

Pi squared (Π²) and 987

Pi squared (Π²) is 9.8696…, which, if you round to 9.87 and ignore the decimals, is 987, the 17th number of the Fibonacci series. (Contributed by William Erman.)

yes as you also can multiply 21 * 13 and add it to 34 * 21 to make 987 which are the number 7th 8th and 9th of the Fibonacci series.

Chris Weismann says

PI = 1.2 PHI^2=3.1416

I found this relationship while studying the work of Rene Schwaller de Lubicz. He had discovered in the measurements of PI and PHI, where both numbers governed it’s PI shape. Over the course of 15 years he measured the Luxor Temple in Egypt and showed that Egypt had a sound knowledge of phi and pi long before the Greeks. Ref his colossal work Temple of Man.

I then found this website as I wanted to delve into the relationship further.

andries says

i noticed that pi x (square root phi)= 4 ( or very close) and you can make a rectangular triangle with sizes phi, (square root phi) and 1, both of a charming simplicity.

safari peter says

From the Giza pyramid one derives the simple relation between pi and phi that is 5/6 of pi minus 1 is phi.

Gary Meisner says

That does provide a close approximation of phi. There are endless variations on that approach though, for instance 4 / square root of phi is close to Pi. They’re all interesting, and many are very insightful and creative, but the most meaningful relationships are those that are exact or that can be related to a geometric construction. See this site’s page on Phi, Pi and the Great Pyramid of Giza for other interesting relationships.

Lucien says

4 (square root of Phi) = 5.088078 which is again close to Oberg and Jonson’s figure above.

Abdu says

Relation of Pi to phi can be

Pi *4/33 =0.3808 = 0.381

1-0.381=.619 =0.618

0.381*2= 0.764

1-0.764= 0.236

Ian Cattell says

I noticed whilst writing a program to draw a ‘flower of life’ that the position of the third (and subsequent) circles, after drawing the first two intersecting circles is related by this formula (forgive me, I’m a computer geek, not a math geek) –

Given circles of radius r at co-ords X1, Y1 and X2, Y2, the X co-ord of the center of the third circle, X3 = X1 + ( r / (Pi * Phi) ) * Pi.

Have I restated an existing and known about relationship?

Interesting site by the way. I’ve written a nice Pascals triangle program that shows the relationship between it and prime numbers that you might be interested in. I’ll upload it to my website if you want to have a look.

Ian Cattell says

More easily stated ( I should have noticed earlier) X3 = X1 + R / Phi

u mad ? says

we noticed that the circumference of the golden circle isn’t pi but phi*pi and we also discovered that the golden eclipse area is pi*phi and that it isn’t pi.

Teh Epik Duq iz comin says

I am doing 6th Grade Math project on the golden ratio, thanks whoever posted this 🙂

Greg says

When compared with pythagoras, the angles are 89.1, 31 and 59.9 degrees. Triangle e, phi and pi. Slightly off the 90, 60, 30.

Why???

Maybe calculated on a base of 11 or 9 and not a base of ten???

Armando Hernandez says

Another trigonometric functinon involving phi is:

2*sin(pi/10) = phi = 0.618033…

notice that pi/10 rad = 18 degrees.

The relation above can be obtained by studying que pentagram or the pythagorean star.

j.Iuliano says

Here is a geometric conversion approximation for pi to phi, exact to ten decimals:

(((4*pi^2)-(pi^3))/2)^(3^-1) = phi + sqrt(10^-13)

This formula shows a counterintuitive result due to the idea that the side areas of a pi cubed structure (4*pi^2) is equivalent to the entire cubic pi volume plus two volume cubes of phi !!?? This equation shows the structure of an elliptic curve:

((phi^3)/(pi^3))-(2/pi)+(1/2) = .00000008013

In the elliptic curve equation, (x^3)+(y^3) = 4*x*y, the xy point, parallel to the x axis tangent point of the elliptic cusp curve generated by the latter equation is, x=1.67894731 , y=2.1165347, can be substituted for x:phi and y:pi, respectively.The interesting thing about surface areas being greater than the sum of cubic volumes is this effect happens to black holes, where 1/4 of the surface area of a black hole contains more information than the entire volume of the black hole!?

George Lambert says

Where did you come up with this reference – or did you devise it yourself.

It is very interesting, The only similar reference I can find on the internet to it

is in this pdf.

https://cims.nyu.edu/~kiryl/Calculus/Section_2.6–Implicit_Differentiation/Implicit_Differentiation.pdf

I link to the reference is posted here in this image.

https://cloudup.com/cjtHpKgbvWx

j.Iuliano says

addendum: y in the elliptic curve is actually rewritten to the fraction 8/3=2.6666…

PANAGIOTIS STEFANIDES says

PLEASE REF. links found in Web Side http://www.stefanides.gr:

http://www.stefanides.gr/Html/QuadCirc.htm

http://www.stefanides.gr/Html/piquad.htm

http://www.stefanides.gr/pdf/GOLDEN_ROOT_SYMMETRIES_OF_GEOMETRIC_FORMS_by_Panagiotis_Stefanides.pdf

Flemming Sejer says

In the section “Trigonometric functions …”, I find the first note incorrect. It should read: “Note: The angle of Pi/5 is 36 degrees, of which there are 10 in a circle or 5 in Pi radians.” and not “Note: The angle of .5 Phi is 36 degrees, …”

Luke Williams says

I noticed that a

near perfect 7 sided figure, a heptagon with an angle of

can be constructed from any arm of the

5 pointed star.

http://i761.photobucket.com/albums/xx255/paldenium/image.jpg1_12.jpg

which seems to reflect the

phi=7/5 (Pi/e)

similarly

http://i761.photobucket.com/albums/xx255/paldenium/image.jpg1_14.jpg

the base cuts the points at 1:phi

1.618018

Thanks to Marcus the Maronite and Hirano

Myrtonos says

The area of the golden ellipse is simply the pi-phi product multiplied by the square of the semi-minor axis. The convergent of this product with 22 continued fraction values is [5; 12, 53, 2, 14, 1, 1, 4, 2, 5, 11, 3, 11, 3, 1, 4, 4, 17, 1, 45, 5, 2].

Jorge Xerxes says

I observed that Pi/2 = atan(1/sqrt(phi)) + atan(sqrt(phi)).

Read about it in…

https://jorgexerxes.wordpress.com/2015/07/16/algumas-relacoes-matematicas-entre-a-constante-pi-a-razao-aurea-e-a-construcao-de-piramides/

Gary Meisner says

That is true, as it is derived from the triangle shown in the article with sides of the sqrt of Phi and 1, with a hypotenuse of Phi

1/sqrt(phi) = 0.786151377757423

sqrt(phi) = 1.272019649514070

atan(1/sqrt(phi)) = 0.666239432492515

atan(sqrt(phi)) = 0.904556894302381

Total = 1.570796326794900

Pi = 3.141592653589790

Pi/2 = 1.570796326794900

Pi and Phi can be also related through trigonometry through angles involving the pentagon that divides a circle in five sections.

Beware though of articles that tell you that Pi * Sqrt (Phi) = 4. That is not true.

Khalfan says

And that is precisely what PANAGIOTIS STEFANIDES

(http://www.goldennumber.net/pi-phi-fibonacci/#comment-1759)

is trying to claim for fame and he has gone to extensive lengths to prove the clever paradox that π = 4/√φ based on kepler triangles and that this erroneous

value of π is the correct one, Along the lines he makes the extraordinary claim that some aliens in fact notified mankind about it.

Read a little about it here but don’t laugh please

:

http://www.veteranstoday.com/2015/10/06/pi2/

Jorge Xerxes says

Hi Gary and Khalfan,

Thanks for replying and for your reading suggestions!

I developed a little bit more and prepared two another posts.

Note that, despite of some Math and Physics, the main purposes are Art and Aesthetics.

https://jorgexerxes.wordpress.com/2015/11/12/orbit-study/

https://jorgexerxes.wordpress.com/2015/11/24/cosmological-study/

Best Wishes!

Dominic says

The angle of 0,5 x 1.618 x (57.3 deg/rad) = 46.36 degrees not 36 degrees.

If we multiply Phi by 2/5 or 0.4, we get:

0.4 x 1.618 x (57.3 deg/rad) = 37.08 deg, which is not 36, but certainly closer.

There is no reason why 360 should be related to Pi or Phi, since it was only picked because it was divisible by so many factors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, and 90).

If we search for inter-relationships of 360 with e, Pi, or Phi, we’ll find some but they will be only coincidental, with no native meaning.

Marsal says

Hi

cos (pi/5) = phi/2

Marsal says

“There is no reason why 360 should be related to Pi or Phi”

cos (pi/5) = phi/2

Gary B Meisner says

And yet it is. The relationship is related to the geometry of triangles which divide a circle into 5 parts. The pentagon is based on this geometry, and all its segments are in golden ratio proportion to the others. See more at http://www.goldennumber.net/geometry/.

mason says

no, it’s not. 360 is a completely arbitrary number. “5” is the number related to Φ… dividing a whole into 5 parts creates Φ ratios. If we as humans had decided that circles have 500 degrees, then the number 100 would be significant. Also, to whomever posted the relationship between pi^2 and 987, this is not significant. Yes, 987 is the 17th number of the fibonacci sequence. Yes ignoring decimal places in base(10), pi^2 is ~ 987. But 100 is an arbitrary number in any other base. I could argue that written in base(12), this no longer matters or holds true.

But in base 12, Φ can not be written as a function of “fives.”

The relationship cos(pi/5)= Φ/2 makes relative sense. Φ is related to the functions 2x and 0.5x (and other binary operations using these functions as parameters). pi/5 in radians represents 1/5 of a semicircle circle, and as such, it makes sense that the cosine of 1/10 a circle is equal to half of Φ. Again, this is not a true relationship between Φ and pi because 2pi is an arbitrarily chosen number which represents the whole of a circle, just like 360 degrees.