The FibPhi 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:
f^{n} = ½[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 onetoone 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.
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.
As these are full screen images from my PC being driven at 1600x1200 32bit 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
(18871920) that contained our old friend
½(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
½(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 doublestrand 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:
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

¼Ö{8 – (Ö5 + 1)[Ö3 + Ö(5  2Ö5)]}

¼Ö{8  2f[Ö3 + Ö(7  4f)]}

cos 3

¼Ö{8 + (Ö5 + 1)[Ö3 + Ö(5  2Ö5)]}

¼Ö{8 + 2f[Ö3 + Ö(7  4f)]}

sin 6

(1/8)(Ö5 + 1){Ö[3(5  2Ö5)] – 1}

¼f{Ö[3(7  4f)]  1}

cos 6

(1/8)(Ö5 + 1)[Ö3 + Ö(5  2Ö5)]

¼f[Ö3 + Ö(7  4f)]

sin 9

Ö(½{1 – ½Ö[½(Ö5 + 5)]})

Ö{½[1 – ½Ö(f + 2)]}

cos 9

Ö(½{1 + ½Ö[½(Ö5 + 5)]})

Ö{½[1 + ½Ö(f + 2)]}

csc 9

(Ö5 + 1)Ö{2 + Ö[½(Ö5 + 5)]}

2fÖ[2 + Ö(f + 2)]

sec 9

(Ö5 + 1)Ö{2  Ö[½(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

¼Ö2(Ö3  1)

¼Ö2(Ö3  1)

cos 15

¼Ö2(Ö3 + 1)

¼Ö2(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

¼(Ö5  1)

1/(2f)

cos 18

½Ö[½(Ö5 + 5)]

½Ö(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

½Ö{2 – Ö[½(5  Ö5)]}

½Ö[2 – Ö(3  f)]

cos 27

½Ö{2 + Ö[½(5  Ö5)]}

½Ö[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

½

½

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

½Ö[½(5  Ö5)]

½Ö(3  f)

cos 36

¼(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.
This page accessed


times.

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