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The Fib-Phi Link Page

My Contributions

As I complete new work, I will present it here for you to peruse.  I hope some of you will find some value in it, at one time or another.  So please, keep coming back.

Powers of Phi

If you have access to the free Adobe Acrobat Reader please take a look at my observations regarding the algebraic representation of the Powers of Phi.  You might have to use your right mouse button on this link and click on "Save Target As..." or "Save Link As...".  Then you can save it to your computer and open it with Adobe Reader.  In this document I describe my discovery of the following relation expressing Phi to the positive integer powers:

fn = [F(n)Ö5 + L(n)],  n ³ 0

Phi Portraits

Having calculated vast numbers of digits of the Golden Ratio I wondered what visuals a pictorial representation would yield.  Assigning each of the digits 0 through 9 a unique color, I started plotting from the center of the canvas working my way outward, in a clockwise square spiral.  The results are interesting to study.  Of course being an irrational number no patterns really exist, but one can stare at the various densities and colors for aeons.  This one-to-one plotting of digits by their corresponding color was one method - another was to plot only those positions where contiguous digits repeated.  The blue series was developed in this fashion.

I present a total of four images here, two from each method.  The colored pair are the digits each assigned a color, the denser one being plotted one pixel per digit, the other utilizes 10 pixel blocks to better see the method.  Note that the denser images utilize around the first 1,000,000 digits of Phi!  The following two thumbnails are linked to their full size counterparts; click on each to view the full image.

Phi Portrait Sparse - Click here for Full Image Phi Portrait Dense - Click here for Full Image

The second pair show occurrences of repeating digits in blue.  I assigned lighter shades of blue as the length of repetition increased.  For example, if there are 2 nines together, you will see two blue pixels (or blocks).  If there are 5 nines together, you will see 5 pixels (or blocks) of a lighter shade of blue.  As you will see what appear to be longer runs in the darker color, that would be where multiple runs of 2 digits neighbor, e.g. 443377.  Below are the two images depicting this analysis.

Phi Repeating Digits Sparse - Click here for Full Image Phi Repeating Digits Dense - Click here for Full Image

As these are full screen images from my PC being driven at 1600x1200 32-bit true color they are large, so please be patient while they load.  Your patience will be rewarded!

Golden Ratio Computation Record Quest

Always my quest is the computation of large numbers of digits of the decimal representation of Phi, an ongoing effort.  I will keep you all up to date here as I (hope to) reach new heights.  I envisioned a new algorithm I now must code.  Potentially I may be able to explore much further than my current 15 million limitation.  Also, waiting in the wings, I have finally laid my hands on the Fast Fourier Transform (FFT) algorithms, but I have not started working through them.  Hopefully sometime this summer I will be able to take that step, which should catapult me to finally be in league with the big players.

The following table depicts my computational results and dates for the records I have been fortunate enough to set.  It is my dream to someday lay claim to besting the record for the Golden Ratio, currently at 1.5 billion.

Record Digits Date Still Standing
Phi 200,000 December, 1986 No
Sqrt(5) 10,000,000 December 20, 1999 Yes

Golden Triangle Inscribed in Golden Ellipse

Please read my derivation of the lengths of the two Golden Triangles that can be inscribed in a Golden Ellipse inscribed in a Golden Rectangle.  Click the image below for egress.

Srinivasa Ramanujan expression for Phi

I noticed an expression by the great Indian mathematician Srinivasa Ramanujan (1887-1920) that contained our old friend (Ö5 + 1).  I manipulated it, without (I hope) messing it up, and came up with an expression for Phi that involves e, Ö's, and pi.  It also has a (Ö5 - 1) term, which I expressed as 1/Phi, but that is just a function of Ö5.  I'm sure Ramanujan knew his original equation was relating the Golden Ratio, e, and pi.  I just rearranged the terms to make it an equality for the Golden Ratio.

Please click on the image below to see a larger format.

The Golden Ratio in DNA

In double-strand deoxyribonucleic acid, there are 10 nucleotide pairs in each revolution.  This makes the angle between them 36 degrees, the same angle as at the star point of the regular pentagonal star, or pentagram, i.e. the Golden Triangle.  I just thought Phi in DNA was neat, as I don't recall reading about it anywhere.

Book List

Please visit my book link page.  I've tried to include most of the books I own and interesting ones I found on Amazon for easy access to purchase and review.  I'll write up some of my reviews as time permits and post them for your perusal.  Please click here for the book link page.  I've featured the five books below due to their aesthetic appeal - click on each to view the full image.

Yahoo Group

Please visit and join my Yahoo Group.  I'll try to keep things interesting but I'd like everyone to contribute to the dialogue, links, and photos sections.  Click here to visit; join straightaway by entering your email address in the box below:
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Phi Screenplay

I'm writing a screenplay to a film I hope to direct.  If there is anyone interested in collaboration please contact me.  The premise of the piece is the study of a man intrigued with the Golden Ratio, so much so he eventually suffers a psychotic breakdown in his pursuit of the beauty and truth in Phi.  My screenplay encompassed many of the elements seen in "Pi - The Movie", "A Beautiful Mind", and "Contact" before those films were created.  There is much I have to contribute to this project that will set it apart as its own piece of art.

Trigonometric Golden Angles

I have worked out some of the algebraic representations for the six basic trigonometric functions.  Some people have presented 9, 18 and 36, in addition to the typical 30, 45, and 60, but I hope I've added some interest in the formulae for the multiples of 3 degrees starting with 3.  Also I have expressed the results in straight algebraic form as well as forms involving Phi.  The following table contains some of the equations, showing the beauty and symmetry of the relationships between the Golden Ratio and trigonometry.  This table is a work in progress; expect more next time you visit.

Function Algebraic Formula Phi Formula
sin 0 0 0
cos 0 1 1
csc 0 ¥ ¥
sec 0 1 1
tan 0 0 0
cot 0 ¥ ¥
sin 3 Ö{8 (Ö5 + 1)[Ö3 + Ö(5 - 2Ö5)]} Ö{8 - 2f[Ö3 + Ö(7 - 4f)]}
cos 3 Ö{8 + (Ö5 + 1)[Ö3 + Ö(5 - 2Ö5)]} Ö{8 + 2f[Ö3 + Ö(7 - 4f)]}
sin 6 (1/8)(Ö5 + 1){Ö[3(5 - 2Ö5)] 1} f{Ö[3(7 - 4f)] - 1}
cos 6 (1/8)(Ö5 + 1)[Ö3 + Ö(5 - 2Ö5)] f[Ö3 + Ö(7 - 4f)]
sin 9 Ö({1 Ö[(Ö5 + 5)]}) Ö{[1 Ö(f + 2)]}
cos 9 Ö({1 + Ö[(Ö5 + 5)]}) Ö{[1 + Ö(f + 2)]}
csc 9 (Ö5 + 1)Ö{2 + Ö[(Ö5 + 5)]} 2fÖ[2 + Ö(f + 2)]
sec 9 (Ö5 + 1)Ö{2 - Ö[(Ö5 + 5)]} 2fÖ[2 - Ö(f + 2)]
tan 9 (Ö5 + 1) - Ö(2Ö5 + 5) 2f - Ö(4f + 3)
cot 9 (Ö5 + 1) + Ö(2Ö5 + 5) 2f + Ö(4f + 3)
sin 15 Ö2(Ö3 - 1) Ö2(Ö3 - 1)
cos 15 Ö2(Ö3 + 1) Ö2(Ö3 + 1)
csc 15 Ö2(Ö3 + 1) Ö2(Ö3 + 1)
sec 15 Ö2(Ö3 - 1) Ö2(Ö3 - 1)
tan 15 2 - Ö3 2 - Ö3
cot 15 2 + Ö3 2 + Ö3
sin 18 (Ö5 - 1) 1/(2f)
cos 18 Ö[(Ö5 + 5)] Ö(f + 2)
csc 18 Ö5 + 1 2f
sec 18 Ö[(2/5)(5 - Ö5)] 2Ö[(1/5)(3 - f)]
tan 18 Ö[1 - (2/5)Ö5] Ö[(1/5)(7 - 4f)]
cot 18 Ö(5 + 2Ö5) Ö(4f + 3)
sin 27 Ö{2 Ö[(5 - Ö5)]} Ö[2 Ö(3 - f)]
cos 27 Ö{2 + Ö[(5 - Ö5)]} Ö[2 + Ö(3 - f)]
csc 27    
sec 27    
tan 27 (Ö5 - 1) - Ö(5 - 2Ö5) 2f - 2 - Ö(7 - 4f)
cot 27 (Ö5 - 1) + Ö(5 - 2Ö5) 2f - 2 + Ö(7 - 4f)
sin 30
cos 30 Ö3/2 Ö3/2
csc 30 2 2
sec 30 2Ö3/3 2Ö3/3
tan 30 Ö3/3 Ö3/3
cot 30 Ö3 Ö3
sin 36 Ö[(5 - Ö5)] Ö(3 - f)
cos 36 (Ö5 + 1) f/2
csc 36 Ö[(2/5)(Ö5 + 5)] 2Ö[(1/5)(f + 2)]
sec 36 Ö5 - 1 2/f
tan 36 Ö(5 - 2Ö5) Ö(7 - 4f)
cot 36 Ö[1 + (2/5)Ö5] Ö[(1/5)(4f + 3)]
sin 45 Ö2/2 Ö2/2
cos 45 Ö2/2 Ö2/2
csc 45 Ö2 Ö2
sec 45 Ö2 Ö2
tan 45 1 1
cot 45 1 1

Please drop me a note to let me know how to improve my tribute to all things Golden.

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Changes last made on: Sunday, September 28, 2003, 23:02:46, PST