I Phi Machine Help

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1. May 20, 2017

colinbeaton1

If you know phi it is about 1.618...=2cos36.
The equations when x=phi which is equal to 0 is x^2-x-1=0.
I took the first derivative squared and the second derivative cubed.
The equation with x=phi is:
[2x-1]^2+2^3=13
Check for yourself, if you fill in phi you get 13.
Anyway, I do not know what to do with this sacred equation.
Is it a toroidial field?
Can I make something with it?
Is it a part of the human body, or something else in the universe?

2. May 20, 2017

Staff: Mentor

$(2 \varphi - 1)^2 = 5$ directly follows from $\varphi = \frac{1+\sqrt 5}{2}$, and adding 23=8 on both sides doesn't change that. It is about as interesting as "5-2=3 because 2+3=5". Correct - so what? That doesn't imply any relevance for anything.

3. May 20, 2017

Staff: Mentor

Why? Since $x=\frac{1}{2}(1 \pm \sqrt{5})$ you can find a lot of quadratic expressions which resolve $\sqrt{5}$ and build something around it.
With which kind of equation? It is as interesting as $x^2=5$ is.
$\phi$ is called the golden ratio and you can find literally thousands of books, articles, related equations and similar about it.
What is a toroidial field? It is a real number, not a field.
Probably nothing that hasn't already been done with it. Even buildings have been constructed according to the golden ratio.
Which machines?
More a part of architecture and eventually the human sense of beauty, but this is debatable. As already mentioned: there are thousands of papers about it.

Edit: Here's good point to start at:
https://en.wikipedia.org/wiki/Golden_ratio#Applications_and_observations

You'll also get beautiful pictures of sunflowers, if you google "golden ratio + nature".

Last edited: May 20, 2017
4. May 21, 2017

colinbeaton1

Thank you.
I do not know what to do with the differential equations.
I was assuming a machine could be made with the first and second differential, with an output max of 13 units.
Would you know how to make a machine with this equation?
I have no idea.
Let me know,thanks

5. May 21, 2017

Staff: Mentor

I'm not sure we do either. You haven't explained what kind of machine you're talking about. Are you referring to functions when you say machines?

In the area of differential equations there are things called operators, that take a function and its derivatives as inputs.

6. May 21, 2017

Staff: Mentor

Can you make a machine out of the equation 2+3=5?
What do you mean if you say "machine"?

7. May 21, 2017

Staff: Mentor

The only thing in this context that comes close to something like a machine are the Fibonacci numbers. This is a sequence, which is build as follows: Start with $1$ and $1$ as the first elements of the sequence and then always add the two last numbers to get the next.
This gives $1,1,2,3,5,8,13,21,34,55,89,144,\ldots$ Now if you build the quotient of two consecutive numbers, that is
$$\frac{1}{1}\, , \,\frac{2}{1}\, , \,\frac{3}{2}\, , \,\frac{5}{3}\, , \,\frac{8}{5}\, , \,\frac{13}{8}\, , \,\frac{21}{13}\, , \,\frac{34}{21}\, , \,\frac{55}{34}\, , \,\frac{89}{55}\, , \,\frac{144}{89}\, , \,\ldots$$
you get a sequence which converges to $\phi = 2\cos 36° = \frac{1+\sqrt{5}}{2}$. There are many formulas with this sequence. Many can be found, e.g. on the Wikipedia page for Fibonacci numbers.