Once you know that

you can apply simple algebra and find the actual value ofphi + 1 = phi * phi

First, combine and rearrange the terms to form a standard quadratic equation:

Not too bad so far.2 phi - phi - 1 = 0

Now we can use the Quadratic Formula to solve for **phi**.

The Quadratic Formula says that when you have a standard quadratic equation, like this:

then you can find the value of2 ax + bx + c = 0

_________ / 2 -b +- \/ b - 4ac x = -------------------- 2a

Going back to our equation for **phi**

we get these values for2 phi - phi - 1 = 0

a = 1 b = -1 c = -1

Plugging into the Quadratic Formula gives:

________________ / 2 -(-1) +- \/ (-1) - 4(1)(-1) phi = ------------------------------- 2(1) _______ 1 +- \/ 1 + 4 = ---------------- 2 ___ 1 +- \/ 5 = ------------ 2

Your pocket calculator, computer, or slide rule can tell you, with varying degrees of accuracy, that the precise value of the square root of 5 is (approximately)

2.2360679774997...

1 + 2.2360679774997... ------------------------ ==> 1.61803398874989... 2(remember a square root has two values, one positive and one negative)and

1 - 2.2360679774997... ------------------------ ==> -0.61803398874989... 2

This is puzzling.

The first number makes sense, because we know that the number we're looking for is bigger than one and greater than zero, that being in the nature of **growth** (things get **larger** when you multiply by **phi**, don't they?). But when you've finished unfolding the Golden Rectangle, and you go **backwards** towards the origin, things are getting **smaller** and moving in the **opposite direction**. That's when you meet this unexpected alter ego of **phi.**

Hmmm... We got here because, if you add one to **phi** you get **phi squared**.
What happens if we **subtract** one from **phi**?
Well, multply by -1 and you've got this same inverted side of **phi**, once again moving backwards and down towards the Cosmic Navel.

I'll leave this question to you:
We know that **phi** somehow embodies both addition and multiplication.
We just looked at subtraction - now, what do you get if you **divide** one by **phi**?

Could it be that those seemingly primal operations of
**addition**, **subtraction**, **multiplication**, and **division** are really just
**four faces** of **phi**?

The Graphical Derivation of **phi**

By the way, if

then what does this equal?2 1 + phi = phi

2 phi + phi = ?

4 5 phi + phi = ?

499 500 phi + phi = ? 3 6 501 Did you guess phi, phi, and phi ?

What we've stumbled onto here is the mathematically perfect parallel of the Fibonacci series we started out with. If you recall, a Fibonacci series is defined by each successive term being the sum of the previous two. So if you start out with one and **phi**, you get:

Or, as we just saw:1, phi, (phi + 1), (phi + (phi + 1)), ((phi + 1) + (phi + (phi + 1))), ...

2 3 4 1, phi, phi, phi, phi,

Each term in these two series is equivalent. For instance:

and so on. Notice that each power of1 = 1 phi = phi 2 phi = (1 + phi) 3 phi = (phi + (1 + phi)) 4 phi = ((1 + phi) + (phi + (1 + phi)))

[ The Golden Mean | The Graphical Derivation of Phi | Derive Phi Exactly | Unfold the Golden Rectangle ]