the golden ratio

is the irrational constant  equation  1.6180...[3]



lines a and b (where a > b) are in the golden ratio if:
(a+b)/a = a/b.[3]

 
...it follows that a rectangle whose
sides are in the golden ratio...

the golden ratio

you can't see me, but i still take up space.  equation  i am watching you.[X]


  lines
but don't worry...
good guy i'm a good guy.[X]

 
 ...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

is

i wish you could see me.  equation  i'm always on the outside.[X]


  lines
and it's lonely here...
but i can see things you can't.[X]

 
 ...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over...

only

sometimes i feel it.  equation  reality bends around me.[X]


  lines
and i know...
i've got so much to share.[X]

 
 ...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over... and over...

part

it's not all wonderful.  equation  a spiral that spins two ways.[X]


  lines
sometimes i lose control...
sometimes that scares me.[X]

 
 ...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over... and over... and over...

of

but things are okay.  equation  a good guy.[X]


  lines
so i am a man.
not a beast.[X]

 
 ...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over... and over... and over...
and over...

my

the key is  equation  a balance.[X]


  lines
between the rage that burns me.
and my love for this world.[X]

 
...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over... and over... and over...
and over... and over

equation

i wish you could know me.  equation  a wish...[X]


  lines
i could know you too.
for something good.[X]

 
...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over... and over... and over...
and over... and over... and over...

.

so we are  equation  together.[X]


  lines
eye to eye.
but i see through you.[X]

 
...it follows that a rectangle whose
sides are in the golden ratio...

...can have a square cut out of it,
and the remaining rectangle is golden...

...over... and over... and over...
and over... and over... and over...
...infinitely.

so what?

so what?

  • it's cool.

so what?

  • it's cool.

  • the ratio shows up in the construction of a regular pentagon.[4] This is how the Greeks discovered it.
    Proof:
    1. pentagon Fact: If you take a regular pentagon, and draw diagonals from one vertex to the two non-adjacent vertices, you end up trisecting the angle corresponding to the vertex you drew the diagonals from.
    2. pentagon Using the fact in (1) we see that angle CAD ~ angle EAD ~ angle BDA ~ angle EDA. Then triangle FDA and triangle EDA share two pairs of congruent angles and side AD. Therefore, triangle FDA = triangle EDA (by Angle-Side-Angle). It follows that side AF = side DF = 1.
    3.   Using the fact in (1) and since angle BFC ~ angle AFD (vertical angles), we see that triangle ADF and triangle BCF are similar.

    So  math , but  math .

    Thus  math ,  math , so  math .

    Solving this quadratic equation gives us one positive value for x (distances must be positive):

    math , and so y is our golden ratio.

so what?

  • it's cool.

  • the ratio shows up in the construction of a regular pentagon.[4] This is how the Greeks discovered it.

  • the ratio is culturally important in art and design

  • some say the ratio shows up everywhere in nature.[2]

  • but what the hell does this have to do with Fibonacci Numbers?

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