Theos-Talk Archives (March 2005 Message tt00118)

<< As to ratios, there are several irrational numbers that deal with 
important aspects of existence. ... A third is the Golden Mean. ... 
All four values have both scientific and metaphysical significance. >>

I'm getting a bit off the topic here, but this discussion reminded me of 
one of my favorite cartoons, which I remember seeing when I was a kid, 
and loved to rediscover in reruns on the Disney channel a few years ago 
(and I was lucky enough to catch it on tape, which is hiding here 
somewhere, but I haven't watched it in over a year or so)....

It happened to be during the very first COLOR episode of Walt Disney 
(all prior episodes were black&white). It included a cartoon titled 
"Donald in Mathmagic Land".

See: 
http://www.amazon.com/exec/obidos/ASIN/6301017129/ref=nosim/recommendations

I particularly remember Donald Duck's professor uncle explaining golden 
rectangles, and showing how when you remove a square from the golden 
rectangle, you are left with another golden rectangle.

Every once in awhile, I'll get out paper and pencil and again figure out 
the golden value. Since the fibonacci sequence (every number is equal to 
the sum of the prior two numbers) includes the golden ratio when you 
take the sequence to infinity, I just imagine three numbers in a decimal 
Fibonnaci sequence using just one unknown, X:

X (a number less than 1)
1
1+X

The third number is equal to the sum of the prior two numbers.

Since the Golden Ratio is equal to the next fibonacci number divided by 
the just previous number, then 1/X must be equal to (1+X)/1, so I just 
solve:

X+1 1
------- = -----
1 X

In this example, we solve for X, then we divide it into 1 to get the 
golden value.

Actually, if I think of the three numbers as X-1, 1 and X, we just solve 
for X.

X 1
------- = -------
1 X-1

Just cross-multiply (X^2 - X = 1) and solve for X.

John

P.S. Donald in MathMagic Land was also fun when he was taught how to use 
angles to play billiards, and other "fun" math topics.