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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* that adheres to this formula;

**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...???**

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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 About these *Tau Ratios*

Is that **-1, -2 & -3** DON'T have solutions...???

*What is up with that...???*

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## 2 comments:

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

1 is the loneliest number there will ever be.

618033988749... is there to keep it companee.

Bird

minister of propagadada

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