Survey posted 2005-05-09 15:20 by Jon.

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Isn't the mathematical constant e the golden ratio?

Nope.

what? that's weird... you answered your own question

And within a minute after the question!

Yeah, after reading the other comments.

I think it is related to e. If I remember correctly, a radius (theta=constant) drawn over an exponential spiral (r=e^theta) will always make angle phi radians on the side of the spiral away from the pole every time it crosses the sprial.

of course they must be related somehow. e is the perfect number in calculus, phi is the perfect number in geometry (flat things with sharp angles) and pi is the perfect number in trigonometry (circles not triangles), and i is the perfect number in complex math... that doesn't really work, but oh well, thats what you get for making stuff up on the spot.

Do you think you can expand
e^(pi i) + 1 = 0
to include phi?

of course!

e^(pi*i)+phi=1/phi

: )

But then you get rid of the zero and make it seem like phi is more important...how bout this:
e^(pi*i)+phi=1*Phi+0

Mathematics is supposed to be elegant, & "1*Phi" is not, so there must be a better (that is, more elegant) equation. Maybe e^(pi*i)+1/Phi+phi^2=0 (& then it even includes 2, which all geeks know is important).

Now all it needs is 42. :)

Here is a better one: e^(pi*i)+1/Phi+phi^(42/(3*7))=0
This includes the important mathematical constants, 42 (the answer to Life, the Universe, & Everything), 3 (to represent the Trinity), & 7 (also important in Christianity & Judaism). Obviously, it looks better written in "textbook"/pretty print fashion.

but then we do have phi twice in all of these... Isn't there a way to make phi show up only once? No, I guess not...

Phi & phi both appear so the equation is especially beautiful. :) Actually, there may be a way to get rid of one of them, but it would probably ruin the layout &/or mess up the other numbers. No powers of Phi are integers.

but phi can be computed with integers.

Infinitely large ones, or infinitely many of them...I do not want to add an infinite sum, infinite product, or limit to the equation.

jvdthwip wrote (9 May 2005, 18:24 GMT:

I'll be the 1`st to answer:

== GOLDEN RATIO ==
(1+sqrt(5))/2, approximately 1.61803, which happens to be the ratio of a diagonal of a pentagon to its side. This constant shows up in many metrical properties of the dodecahedron and icosahedron just as the square root of 2 shows up in the metrical properties of the cube. A golden rectangle has sides in this ratio. A golden rhombus has diagonals in this ratio.

So all we have to do is find out how to involve that in the equation...

Can anybody solve that for Phi, so we can get a formula for Phi in terms of e?

I assume he meant e^(pi*i)+Phi=1/phi (or with the (P/p)hi's reversed, which would be the same thing), which yields 0=0 upon attempt at solving for Phi, so, unfortunately, no.

No.. the equation is e^(pi*i)+Phi=1/Phi [both with capital letters].

I can get to:
Phi^2+Phi*e^(pi*i)-1=0

-e^(pi*i) + sqrt(2e^(pi*i)+4)
------------------------------ = Phi
2

That looks way too nasty though..

sqrt(2e^(pi*i)+4) - e^(pi*i)
----------------------------- = Phi
2

Or all on one line as:

Phi = .5 (sqrt(2e^(pi*i)+4) - e^(pi*i))

Dammit.. It dont work.. Oh well.. I tried.

You are right, his equation does work as e^(pi*i)+Phi=phi (or with phi replaced with 1/Phi, of course). Sorry & thanks for the correction.

You are right, his equation does work as e^(pi*i)+Phi=phi (or with phi replaced with 1/Phi, of course). Sorry & thanks for the correction.

Oops...double post.