The golden ratio, or divine proportion, or PHI
, is simply a number, nearly 1.61803399
, and its discovery is attributed to the Greeks. What is astonishing is the frequency with which the number appears in art, music, and even nature. The appeal of the golden ratio to human psychology has been scientifically tested, beginning with German psychologist Fechner and followed by several others.
But objects constructed with the golden ratio in mind are not just pretty to look at; the mathematical properties of this elusive number are just as interesting. I’ll describe some of the more simple properties here.
Derivation of Phi
In my opinion, the easiest derivation for the golden ratio is to use the Golden Section definition:
The Golden Section is the division of a given unit of length into two parts such that the ratio of the longer to the shorter equals the ratio of the whole to the longer.
Thus, if we take a unit line and let x
be the longer part and call the corresponding shorter part 1x
, we obtain the expression for the Golden Section:
[Unparseable or potentially dangerous latex formula. Error 4 ]
Substitution yields:
[Unparseable or potentially dangerous latex formula. Error 4 ]
Trivially, we use the quadratic equation to solve, and take the positive root (since it is defined as such by the Golden Section):
[Unparseable or potentially dangerous latex formula. Error 4 ]
Note: it’s actually easier to solve for x
and then take the reciprocal, but then many of the curious identities of the golden ratio remain hidden.
Peculiar Properties
In the course of the derivation, the following peculiar properties have emerged (for brevity, proofs have been omitted):

To find PHIsquared
, simply add 1 to PHI
. Consequently,
for any n
:
[Unparseable or potentially dangerous latex formula. Error 4 ]

The difference of PHI
and its reciprocal is the whole number one.
And in general:

Construct a series, where, for any even integer n
:
[Unparseable or potentially dangerous latex formula. Error 4 ]
And for any odd integer n
:
[Unparseable or potentially dangerous latex formula. Error 4 ]
That is, for any n
, you always get a whole number.
Take the ratio of any two successive numbers in this series, and
its value converges to PHI
!
The Fibonacci Numbers
The Fibonacci numbers form an interesting sequence defined recursively by:
[Unparseable or potentially dangerous latex formula. Error 4 ]
Like PHI
, the Fibonacci numbers are abundant in nature, so you might reason that the Fibonacci numbers and PHI
are related in some way. And you’d be correct. If we take the ratio of any two successive elements, for example:
[Unparseable or potentially dangerous latex formula. Error 4 ]
we find that the ratio, much like the PHI
sequence, converges to the value of PHI
!
Conclusion
All this just to say that the photographs from my recent Australasia trip are now available online. They are formatted using a 9:6
aspect ratio, which is as close to the golden ratio as we can get using traditional photography. I hope you enjoy them.