1. Reflections on the Gold, Silver
and Metallic Ratios
Numbers that have Accompanied and
Fascinated Man for Centuries
Dann Passoja
New York, NewYork
Winter 2015

2. 2
My Recent Involvement with Two Exceptional Numbers
Φ and σ
the Gold and Silver Ratios
I'm a reader and I've always enjoyed reading books of all kinds. With the
advent of the internet, people's intellectual consumptions probably won't change
too much but the need to read books probably will because a great deal
publishing will be done via the internet. That will never satisfy my need to get
information directly from real books though. Presently my information
consumption has become divided between books and virtual books (their
electronic representations)
I became interested in the gold and silver ratios from two different
sources: Huntley's book "The Divine Proportion" and the work I was doing on the
"Passoja- Lakatakia Set". Both of these interests came about as a result of my
work on fractals.
I was reading Huntley's book on the Divine Proportion book one day in
about 1979 (or so) and I found a discussion about Pascal's triangle on page 135.
I had become familiarized with fractals after reading B.B.Mandelbrot's on
"Fractals, Form, Chance and Dimension" and one figure in it, namely
"Seripinsky's Gasket" seemed to apply to the observation Huntley had made,
namely that multiples of certain integers on Pascal's Triangle create triangular
forms.
Once I got home and pulled out my hand calculator It didn't take very long
find that, yes, indeed, it was part of the Seripinsky Gasket. I needed a large
computer in order to explore some of these issues further. As usual my Dad was
extraordinary and must've anticipated my needs because he left some money in
his estate and I was able to proceed in my quest to find fractals on Pascal's
triangle thanks to him. I was able to purchase a computer work station and
installed it in my apartment in NYC.
I was a full time artist at this point and I knew my curiosities in both art and
mathematics would eat up a great deal of my time. So I worked a double shift:
paint during the day and do mathematics at night. I painted some wonderful
pictures (an artist considers his pictures as his children) and made some exciting
mathematical discoveries at night.
My need for exploring art, science and mathematics has never diminished
my need to search, explore and create, but getting a good night's sleep once in a
while often has priority over my explorations.
Nevertheless, explore, I did. Soon, Art and Science fused in me and I
experienced peacefulness and finally felt comfortable about something that had

3. 3
always bothered me, namely, that Art and Science had become disconnected in
people's minds.
The General Field of Explorations in this Work
There's some elementary geometry and some elementary mathematics
that I want to forward in the following work. I've forgone rigor and replaced it with
an operational approach to everything. My thoughts are quite straightforward on
this : the best way to explore mathematical ideas is to present the problem, "fool
around with things" (doing some calculations if you want to) others, if they're like
Gauss, can do extraordinary things in their heads and so they obviously will
dispense with such matters.
My work here is elementary and transparent. I hope to present something
for the reader to think about. He/She should be able to understand everything at
first glance. This is not "High Level Mathematics" simply because I didn't want it
to be.
My interest is one of fascination too. I consider myself a mathematician of
sorts more interested in the use of mathematics in Physics and computers than
in almost any other feature of math, nevertheless that certainly doesn't quell my
interest and fascination in math. I usually work out mathematical things that
interest me with a great deal of effort and lost sleep. However, that doesn't stop
me in my quest to learn more and more mathematics because I really like
mathematics. My approach is that I really like it, enjoy learning it, enjoy using it
so I plunge in, make out an activity list, define some goals and come up with a
schedule.
The deficiencies in my quest are in finding good resources. It takes quite
some time to find readable and understandable information. Too many times I
just say to myself "this is just pathetic". Fortunately there are usually one or two
people that stand above all the rest and bring light onto an item. I cite for
example Richard Feynman his contributions to our scientific knowledge and his
teachings are readable and just wonderful. Einstein once said
" If you can't describe things to a 6th grade
student then you don't understand it"
In this work I've followed Einstein's comments.
The Golden Ratio, the Silver Ratio and the Metallic Ratios
The golden ratio, φ and the silver ratio σ are just two numbers that have
survived antiquity and the metallic ratio µ is a generalization on these types of
numbers.
There are also a geometric forms that can be related to these numbers.
One of which is shown below:

4. 4
These are some spiral constructions made from within rectangles that
have ratios of their sides = φ. I put this Figure here because it demonstrates the
use of φ and to introduce the reader to the manifestation of φ as a hidden object.
It is hidden in a geometrical construction.
I've also included two examples of Golden Ratio spirals taken from a
nautilus and from a sunflower:

5. 5
The nautilus has evolved over millions of years and the present day
nautilus closely resembles its ancestors. The chambers contain the animal at
different stages of its growth cycles.
It's possible to find evidence of intertwined spirals in the center of a
sunflower. Sunflower seeds come from this place and are eaten by many of us.
The center of a sunflower blossom
Computer reproduction of the central part of a sunflower blossom.

6. 6
I'm including the following article as evidence of how ubiquitous the
Golden Ratio can be in the minds of men. I consider this article to be interesting
but that's all it is because a substantive analysis of the observation is missing. It's
a nice observation though.

7. 7
There has been a plethora of work done on Phi both geometrically and
mathematically, over quite a long period of time, starting from the time of the
Grecian Empire. Phi's attraction to everyone is based, in part, to its simple and
astonishing usefulness, its pervasive appearance in Nature's realm and its ability
to be a fundamental influence in the design of man's structures.
The Character of φ, the Golden Ratio
Phi is just a number having some interesting mathematical properties- it's
an irrational number. I always remember that φ is:
φ =
1+ 5
2
= 1.618033988...
having some important mathematical relationships such as:
1
φ
=
5 −1
2
= 0.618033988...
and one that seems trivial but is not:
φ −
1
φ
= 1.000000...
I present below both geometric mathematical realizations of φ for comparison
purposes:

8. 8
Phi has an amazing continued fraction expansion :
(2
and
(3
The Character of σ, the Silver Ratio
The silver ratio σ is another interesting number that also has an interesting
history.
(4
It is similar to φ since:
And has a continued fraction expansion of:

9. 9
The Character of the Metallic Mean
The metallic mean is a mathematical generalization of the golden and
silver means. It is:
µ =
n + 4 + n2
2
and has a continued fraction expansion of:
Which generates the following set of numbers:

10. 10
which all have the same properties of the golden and silver means.
Mathematics and Properties
I have found that the mathematical properties of φ, σ and µ are quite
interesting and I'd like to list some examples below because I haven't been able
to find these statements elsewhere. I suspect that they've been "kicked around"
before.
I'll explore some calculus and series expansions in order to bring in a
different point of view. I've found that some of this content to be provocative,
feeling that it demonstrated something that I should know. In fact I did know most
of the statements but they often had an impact in the form of "...I already knew
this"... but it seemed that using Phi helped me gain insight about a great number
of things that I already knew.
Properties of Phi

11. 11
I'll be using φ in all these examples because all of the other special
numbers follow suit.
Some of these things are known, others are not
Relationships for Phi and Powers of Phi
1.
2. For terms based on φn
+
1
φn
the behavior of the odd and even powers of
m are different. The terms are:
For odd powers f(φ)= 5 × 1,2,5,13,34,89,233,610....( )
For even powers, m f φ( )= 3,7,10,18,47,123...
The series of numbers in the odd powered series has a recursion
a n( )= 3a n −1( )− a n − 2( )
and the even powered series is a bisection of the Lucas numbers
a n( )= L(2*n)
Where L(n) are the Lucas numbers (closely related to the Fibbonacci
series). The Lucas numbers are L n( )= L n −1( )+ L n − 2( )
L n( )= 2, 1, 3, 4, 7, 11, 29, 47, ... The Lucas sequence is
related to the Golden Ratio by:
Power of Phi φm
φ−m Fractional Parts
1 1.6180933... 0.6180933... *****
2 2.6180933... 0.38196601... 3
3 4.2360679... 0.236067977... *****
4 6.854101... 0.145898033... 7
5 11.0901699... 0.0901699... ****
6 17.94427... 0.05572809000... 18
7 29.03444185... 0.0344418537... ****
Summation
n=0
m
∑ φn
+
1
φn
= φm
( ) 5 + 3( )+
1
φm
⎛
⎝⎜
⎞
⎠⎟ 5 +1( )
with:
f φ( )m+1
f φ( )n
= φ
φφ
= φ −1( )−φ
Ln =
φn
− 1−φ( )n
5

12. 12
φn
= φm
φ +1( )
n=0
m
∑ −φ
1
φn
n=1
∞
∑ = φ
For the odd powered series φ j
= φm
φ − φ +1( )
j=2n−1
m
∑
The inverse of Phi to an odd power equals the fractional part of that
number.
The whole number part of Phiodd power
is a sequence 4, 11, 29, 76, 199, 521,
1364, 3571,... which is a bisection of a Lucas sequence
The Lucas sequence is 2, 1, 4, 7, 11,18, 29, 76, 199, 521, 1364, 3571,
9349, 24476...the ratio Ln/Ln-1→φ+1
The Lucas sequence is related to the Golden Ratio by: Ln =
φn
− 1−φ( )n
5
1
φnφ
=
1
1−φ−φ
n=0
∞
∑
1
nφ
=
HarmonicNumber[m]
φn=1
m
∑
1
φ2πin
=
−1+ φ( )−2i 1+m( )n
−1+ φ( )−2iπ
n=0
m
∑
Product Series
φn
= φm
n=1
m
∏ φm2
φn
( )
n
n=0
m
∏ = φ
1
6
m m+1( ) 1+2m( )
1
φn
⎛
⎝⎜
⎞
⎠⎟
n
n=0
m
∏ =
1
φ
⎛
⎝⎜
⎞
⎠⎟
1
2
m m+1( )
φφ
= φ −1( )−φ
1
φ j
= φ −φ φ −1( )
j=2n−1
m
∑

13. 13
Calculus - some derivatives
df (φ)
dx
and integrals f (φ)dx∫
Derivatives
1.
d
dx
φx
( )= ln φ( )φx
2.
d
dx
xφ
( )= φx
1
φ
3.
d2
xφ
dx2
= xφ−2
4.
d
dn
1
φn
⎛
⎝⎜
⎞
⎠⎟
n=1
∞
∑ = φ ln
1
φ
⎛
⎝⎜
⎞
⎠⎟
5.
dxφ
dφ
= xφ
ln x( )
Integrals
1. φx
dx =
φx
ln φ( )∫ = 2.07808692... 1.618033...( )x
2. xφ
dφ =
φ φx
2 x +1( )∫
3.
1
φx
dx =
1
φ
⎛
⎝⎜
⎞
⎠⎟
ln
1
φ
⎛
⎝⎜
⎞
⎠⎟
∫
.
Consideration of φ as a base such as "e" and 10.

14. 14

15. 15
This is something that is possible but it doesn't seem to be an improvement on
the bases 10 and "e". The reason that it should be considered comes from the
fact that our decimal number system is based on whole numbers and fractions ie
x +
1
y
= XXX.yyyy...
for example and
φ = 1+
1
φ
might represent this in similar terms.