Is the "golden ratio" related to "e"?

Pretty much the subject line says it all.

I remember from my physics courses all the places that e, the base of the natural logarithm, showed up in nature & natural processes. From memory, it’s 2.718281828459045… (there’s a trick to remembering it*).

Now, thanks to The DaVinci Code everybody is talking about Phi, the so-called “golden ratio”, roughly 1.618.

Are these two values related in same way?


*the trick, by the way is “2.7”, Andrew Jackson squared, triangle.
“2.7” - you just have to remember that.
“1828, 1828” is Andrew Jackson’s election year, twice
“45, 90, 45” angles in a 45-degree right triangle.

Yes, I am a geek. Why do you ask?

The golden ratio is an irrational number, roughly 1.618…

For any A and B where A > B, if the ratio of (A+B) to A is equal to the ratio of A to B, then it is the golden ratio. I don’t think it’s directly related to e in any way, but I’m no math nerd.

No more than any other two numbers. phi is algebraic, which means that it can be expressed as the solution to an equation which uses only rational numbers and certain “basic” operations. Or more specifically, phi = (sqrt(5) +1)/2. e, on the other hand, is transcendental, which means that it is not the solution of any such equation. The vast majority of numbers, incidentally, are transcendental, but most of them aren’t too interesting for practical purposes. Probably the only transcendental numbers you’re familiar with are e and pi (and things derived from them, like 2pi, and e[sup]2[/sup], and so on).

BTW, Wikipedia has a pretty good page on the golden ratio.

e = sum((2[symbol]f[/symbol] - sqrt(5))[sup]n[/sup]/n!, 0 < n).

Ultrafilter, you’re workin’ too hard, and the result isn’t really germane to phi.

How about this, deriving from the properties of the Golden Rectangle:
(1/2) ((e^(i36Pi/180))+(e^(-i36Pi/180)))=phi/2
(that is, the cosine of 36 degrees is Phi/2)

You’re both working too hard. e = φ • 0 + e

(More seriously, those are both interesting relations.)

There’s another good one at Mathworld. Among the mathematical facts on that page is this amusing observation:

Perhaps that could be called the Fool’s Golden Ratio.

Cal’s is good. Mine may not be as interesting as you think…

There is an obscure connection due to the Fibonacci sequence.

Given two consecutive Fibonacci numbers F(k) and F(k+1), the ratio F(k+1)/F(k) tends toward phi as k->INF.

It is also possible to prove (knowing the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, …):

e = --------------------------------------------------

This Page details the proof, which relies on a knowledge of operator algebra (the operator H in this proof is such that Hy(x) = y(x+1)-y(x). The operator symbol “e^D” indicates the familiar derivative operator d/dx is to be plugged into the Taylor expansion for e^x).

Ah ha!

You took the relation e = sum(1 / n!); then re-arranged φ = (1 + √5)/2 into 1 = 2φ – √5; then substituted for 1 in the first relation.

Why you’d go to Wikipedia or other sources is beyond me.
Is phi a mystical number as claimed in The Da Vinci Code?

Well, this actually doesn’t have much to do with e as the base of the natural logarithm. It’s more to do with the “exponential map”, which in local coordinates tends to look like the MacLaurin expansion of e[sup]x[/sup] and is commonly (and abusively) denoted by e[sup]x[/sup] outside of the context of real numbers where that notion really applies.

(1 - e^(x^ipi))(x^0.5 - 1) is related to phi somehow.

Just look at the Newton Basins for the equation. Definitely some golden ratios in that fractal.

Here is a link to an image of the equation’s newton basins centered at 0 between 3 and -3 in the y direction and 3i and -3i in the x direction in the complex plane.