What are the E, pi, phi constants relations

[jcsd]

Nope, there just identities of cos x and sin x.

 

[Hurkyl]

The hyperbolic and circular trig functions are related through complex numbers. E.G.

cosh ix = cos x
sinh ix = i sin x

 

[jcsd]

The hyperbolic functions do have simlair identities though:

sinh x = ½(e^x - e^-x)

cosh x = ½(e^x + e^-x)

e^x = cosh + sinh x

 

[synergy]

No, lavalamp, I meant phi+(phi)^2 = phi^3, and it is the only number that has this property. Phi is one of the roots to y=x^2-x-1 and so x^2=x+1, x^3=x^2+x, etc. It's rather a cool property. Start with phi=(1+root(5))/2 and construct a fibonacci sequence with 1 and phi as your starting numbers. Your sequence will be 1,phi, 1+phi, 1+2phi, 2+3phi, 3+5phi, etc.
Your sequence will also be phi^0, phi^1, phi^2, phi^3, etc.
so phi+phi^2=phi^3
Aaron

 

[lavalamp]

Weird, I've never come across that before. I've never even thought about that before.

 

[Hurkyl]

I meant phi+(phi)^2 = phi^3, and it is the only number that has this property.

Actually, three numbers have that property...

0, φ, and (1-φ)

 

[HallsofIvy]

Dang, Hurkyl, you got in before me!

 

[synergy]

Actually, I knew about (1-phi), I meant to say that the roots of the equation x^2-x-1 were the only numbers with that property, only I didn't think of zero (I added a root when I multiplied that eqn by x).
Aaron

 

[lavalamp]

I assume that you guys are American, since HallsofIvy & Hurkyl effectively don't have profiles.
So I want to know, at what age did you guys learn about the golden ratio, the sin, sinh, cos, and cosh identities and the e^i[pi] thing? And did you learn it in school or just reading in and around Maths?

 

[jcsd]

Originally posted by lavalamp
I assume that you guys are American, since HallsofIvy & Hurkyl effectively don't have profiles.
So I want to know, at what age did you guys learn about the golden ratio, the sin, sinh, cos, and cosh identities and the e^i[pi] thing? And did you learn it in school or just reading in and around Maths?

Well I'm English, I took maths, physics, chemistry and further maths as A-levels before going onto to physics at university. I didn't learn any of this stuff at A-level (They didn't even teach us imaginery numbers or hyperbolic functions at A-level)

 

[synergy]

golden ratio and fibonacci stuff: reading math stuff as a child.
rest of it: a little in high school and mostly in college.
I figured out the stuff about phi+phi^2=phi^3 on my own, though.
Aaron

 

[lavalamp]

I must say, I've never really wondered when x^2 + x = x^3 myself. But then I've never really read around maths or physics before now, so this stuff is all kind of new to me. I'm just sending out a few feelers to see what situation other people are in.

I'm at A Level now, I've just dropped further maths, but I've kept math, physics and chemistry. The very next day after I dropped further maths, they (the teacher and one other pupil), went on to do P5 and the first thing they did was hyperbolic trig ratios. Shame really because I really wanted to learn about those, I'll just have to borrow his notes.
I'm hoping to do aeronautics next year at University.

 

[MathematicalPhysicist]

Originally posted by HallsofIvy
Well, they are real numbers! Any other relationship I suspect is more "number mysticism" than mathematics. (Phi, in any case, is an algebraic number while e and pi are not.)

i found that there is an approximation that 4/(phi)^0.5=pi.
i checked in my calculator and it's precise only from 2 place after the point of a decimal.

i stumbled upon this equation in this webpage:http://www.innerx.net/personal/tsmith/Gpyr.html

is it a reliable approximation?

 

[MathematicalPhysicist]

Originally posted by loop quantum gravity
i found that there is an approximation that 4/(phi)^0.5=pi.
i checked in my calculator and it's precise only from 2 place after the point of a decimal.

i stumbled upon this equation in this webpage:http://www.innerx.net/personal/tsmith/Gpyr.html

is it a reliable approximation?

no one can say anything about this approximation?

 

[HallsofIvy]

What do you mean "is it a reliable approximation"? You, yourself, said that it was correct only to two decimal places.

To 9 decimal places 1/√(φ)= 3.144605511 while π= 3.141592654. The error is 0.003012857. That's precisely how reliable it is!

 

[Bob C]

Relation between pi, phi and e

Just came across this thread and happen to be discussing the same thing in the Google sci.math forum:

If you list these numbers out to 12 decimal places along with the Fibonacci values and the sequence for these numbers (I start the sequence with zero), you will see that the 12th decimal place is the first position where all digits are equal (9). And at this point the Fibonacci value is the square of its position in the sequence. Probably means nothing... just another interesting relation with these numbers.

col 1 is the Fibonacci value, col 2 is the sequence, col 3 is pi, col 4 is PHI, Col 5 is e:

Fib Sequence pi PHI e
Val
0 0 3 1 2
1 1 1 6 7
1 2 4 1 1
2 3 1 8 8
3 4 5 0 2
5 5 9 3 8
8 6 2 3 1
13 7 6 9 8
21 8 5 8 2
34 9 3 8 8
55 10 5 7 4
89 11 8 4 5
144 12 9 9 9 *****

 

[moshek]

pi ^ (phi^2)=pi * pi ^( phi)

 

[MathematicalPhysicist]

pi ^ (phi^2)=pi * pi ^( phi)

this feature can be attribute to any base (different than 1 and 0) because it is simply pi^(phi^2)=pi^(phi+1) and then phi^2-Phi-1=0 which is ofcourse the equation for calculating the solution to the golden number, it doesn't say they have any relations which can only be attributed to these number only.

 

[KingNothing]

Is that golden ratio constant the actual number, or an approximation?

 

[moshek]

Than you should reply to me
in some different way like:

Now i have more exact way
to ask my original question !

Best
Moshek

I invite you to read:

www.physicsforums.com/showthread.php?t=17243

 

[HallsofIvy]

Is that golden ratio constant the actual number, or an approximation?

The golden ratio constant is the actual number. In fact, it is exactly
$$\frac{1+\sqrt{5}}{2}$$.

I can't imagine any mathematical constant being an "approximation to an actual number"!

 

[moshek]

It was Hipasus of metapontum who discover the existence of irrational numbers. he find self similarity on the pentagon so he prove by this that the golden ratio is not a rational number . But for some reasoned Euclid wrote on his 10 th book only about the irrationality of the root of 2 by the classical prove but this Euclid make a confusion with what really happened
with irrational numbers.


Moshek

 

[MathematicalPhysicist]

Than you should reply to me
in some different way like:

Now i have more exact way
to ask my original question !

Best
Moshek

I invite you to read:

www.physicsforums.com/showthread.php?t=17243

i think i did it in first two pages of this thread.

 

[moshek]

I look but i did not see that you did it
anyhow it is really not the main point here.

Moshek

 

[epii10]

equation of unity involving Pi, e, Phi

Thought you guys might appreciate a little math I found concerning the numbers; Pi, e, and Phi as they relate to three unique right triangles. I call it the "Triple-Triangle-Theory" (TTT). I'm no mathematician so if anyone could tell me "why" the TTT works please feel free. (I think it has to do with the imaginary unit ie. "sqrt -1").

Here's the link:

http://www.gizapyramid.com/rick_howard research.htm

Rick Howard

 

[lavalamp]

That's a dead link.

Edit: Just try pasting the URL into the page, the forum should make it into a link automatically.

 

[MathematicalPhysicist]

Thought you guys might appreciate a little math I found concerning the numbers; Pi, e, and Phi as they relate to three unique right triangles. I call it the "Triple-Triangle-Theory" (TTT). I'm no mathematician so if anyone could tell me "why" the TTT works please feel free. (I think it has to do with the imaginary unit ie. "sqrt -1").

Here's the link:

http://www.gizapyramid.com/rick_howard research.htm

Rick Howard

he doesn't say how he arrived at the e proportion.
can someone tell me about it?
the pi and phi proportions are clear to me.

 

[epii10]

e proportion

Loop,

I found the e proportion just messing around with the angles of the Pyramid with my HP graphic calculator. Sort of an accident really, only (and this is going to sound weird) I loved the number e and would talk about it to anyone who would listen and had it in the back of my mind that I would find it in the Pyramid. Boy was I surprised!

My TTT paper (linked above) goes over the derivation of the e proportion but the first paper I wrote before discovering the TTT was all about the e proportion.. here is that link:

http://www.gizapyramid.com/ricks-e-proportion/rick-howards-research.html

I once had a real mathematical guy analyze the TTT and he said it was just a trick through the sustitutions.. (again, I'm no mathemetician) but simply because the equation embodies all three nearly congruent angles, and that coupled with the tetrahedral angle of 60 degrees being the only other solution to a specific question I pose along the way such that the output remains in the realm of "real" numbers, I tend to think it's just a little weird and still can't wrap my mind around it.

Pretty cool huh?

Rick

 

[MathematicalPhysicist]

howard have you tried plugging other numbers for the base of the triangle other than 1?

perhaps it's just a mistake but i notice that when the base is 1 then the side of the square (the base of the pyramid) is 2 and this number is in the proportion of e: beta/theta=e/2 perhaps there is a connection between the side of the base of the pyramid square and this proportion (and perhaps I am hallucinating who knows (-: ).

 

[matt grime]

there's nothing special at all in those calculations, and the thesis contains many unjustified statements.

1) what does "optimal" mean, other than "supports me theory"
2) why do you say irrational numbers are impossible to construct with any accuracy in the real world, with the implicit assumption that somehow rationals are constructible to some degree of accuracy, and if so that negates your statement as then phi is realizable as a diagonal of a pentagon?
3) The mathematical presentation of the paper is shockingly bad, such as anyone who looks at the first diagram is given the impression that the betas are all the same when they certainly aren't.
4) why on Earth is 2 so special to the point where you say no other integer will do it better?
5) the idea that you're going to produce a polynomial with phi as a root and 'wth the minimal number of terms' must be considered an ill advised boast since I'm sure i can think of one of degree 1 with phi as a root, and two if you must make me have integer coeffs, which must have fewer terms than the one you derive (a mathematically provable fact), though i didn't see it highlighted later in the text, but then that's because there's so much unnecessary waffle that it's not an easy task to extract information from the article.
6)the ideas of eqn 5.4 indicate a lack of understanding about raising complex numbers to exponents; again nothing special is going on there.
7) it would be beneficial to learn about polar representations of complex numbers to see why these coincedences occur.