## Saturday, June 18, 2005

### Alternative NotSoGolden Ratios...!

If you've read that terrible book 'The Da Vinci Code' recently,
You may recall that within it's drab & characterless pages,
Dan Brown considered The Golden Ratio

for a few if those sheets...
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As a quick reminder:
The Golden Ratio is a rather Magickal Number

1 / x = 1 + x
Where x = .618033988749 ( and then some more digits... )
1 / .6180 = 1.6180 = 1 + .6180 = 1.6180
Isn't that clever...!!!
What is somewhat less well known--
Is that there is another x that satisfies this same Expression...!
It is -1.618033988749...
That doesn't look so Very Surprising does it...
But in a moment, it will look more Surprising...! ( ? )
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What if you took The 1 in the above Expression
And changed it to some other number...???
Like a... uhhh... oh-- a 2!
What would you get?
The solution for x would then be: .732050807569...
And it's AntiSolution would be: -2.732050807569...

[ i call these alternative Base Ratios; "Tau Ratios" ]

This still isn't Very Surprising... ( is it? )
But there are solution for all of The Positive Integers...
And All of The Positive Real Numbers ( with fractions )
If The 1 ( or 2 ) is replaced with a .5 ...
The Solution for x is: .5 & -1
That's a little bit more Surprising...???
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But then;
What if we try Negative Numbers...!!!
Then it becomes VERY Surprising...!!!
...Because while there are

NO SOLUTIONS for -1, -2 & -3...
There is a Solution for -4...
It's -- 2 ...!
-4 /2 = -2 & -4 + 2 = -2
This is the Only time where x is a Whole Number...!!!
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-5 has a Very Different kind of Solution,
As do all the other Negative Numbers...
-5's Solution for x is: 1.38196601125 & 3.61803398875
-5 / 1.3819... = -3.6180... & -5 + 1.3819... = -3.6180...
-5 / 3.6180... = -1.3819... & -5 + 3.6180... = -1.3819...
Look at that...!
Not only are the two solutions

Very different looking from each other,
But the Equalities are Reflections of One Another,
Rather than Inverses...! ( ? )
And The AntiSolution contains The Golden Ratio...!!!
This is Very Unusual...
Or is it...???
---
The Solution for x when -6 is used...
Is: 1.26794919243 & 4.73205080757
The AntiSolution contains The 'Tau' Ratio for '2'...!
( .73205080757 ) !!!
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For -7; x = 1.20871215252 & 5.79128784748
Guess what...
.79128784748 is the Tau Ratio for '3'...!!!
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What i think is The Most Interesting

Is that -1, -2 & -3 DON'T have solutions...???
What is up with that...???
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Anonymous said...

Hi all.

Solving l/x=l+x you get the twi solutions:

x=-(l+sqrt(l^2+4*l))/2

and

x=(-l+sqrt(l^2+4*l))/2

There are no *real* solutions for l=-1, l=-2, and l=-3 because the sum l^2+4*l is negative in these cases. (Not so golden ratios get rather complex then ;-))

The solution for l=-4 is rather easy because then l^2+4*l=0 and so we have:

x=-(-4+sqrt(0))/2=2

and

x=(-4+sqrt(0))/2=-2

The equation l/x=l+x can be rewritten as x^2+l*x-l=0, where x is different than 0. From this you can easily see why the "equalities are reflections of each other", as you say.

No magic numbers. No ghosts and spirits. Plain simple crystal clear mathematics.

Sincerely yours,
Nick

Bird said...

1 is the loneliest number there will ever be.
618033988749... is there to keep it companee.

Bird