The Golden Ratio
ABSTRACT
The Lorentz transformation (if x = ct) is the same the golden ratio: .

1. Introduction

So, the aim of this work is to derive the golden ratio connected to Lorentz trans- formation. Usually the golden ratio is derived from the a rectangle  but book  derives the golden ratio from a right-angled triangle. This makes possible to connect Lorentz transformation with the golden ratio. The Lorentz factor formula can be the easiest explained from the right-angled triangle. The formula is in fact the formula of one of the right-angled triangle side:

1) It can be seen from this formula also that “c” is the hypotenuse of the right-angled triangle. At the same time “ ” is one criterion of Lorentz factor formula as well. That means, that the hypotenuse of the right-angled triangle is “c”, the other sides is “v” and . This allows that from so defined right-angled triangle to explain the formula of Lorentz transformation too and the golden ratio formula as well. For the derivation of the equations high school mathematics is used. That’s the reason why the equations are more transparent and easier to understand because they are simply.

2. We Introduce First the (t'/t) Formula

The derivation of Lorentz transformation can be found at the end of the book marked by  in the Appendix I. The derivation of Lorentz transformation equations starts from the fact that we measure two light-rays move face each other, in a standing and in a moving coordinate system. So the travelled distance of the light-ray in the standing coordinate system is . Naturally in the movable coordinate system the travelled distance of the light-ray is . Here it can be seen that Lorentz’s starting point were a few assumptions, (see  Appendix I) and from these simple assumptions got the formula named after him. After that Einstein substituted in the Lorentz trans- formation equation , the equation of . This derivation can be found in the book marked number  on the page 32. Now I show this derivation.

2) So, and : 3) Equation (t'/t): We are going to prove that the equation 3) is the golden ratio.

2.1. Now We Derive the Other Form of the Equation (3)

Before we start with the derivation of the golden ratio, we transform the Equation (3).

4) In further derivation we shall use the following equations from the Appendix I of the book  :

5) In further derivation we shall use the following equations from the  :

( See the Figure 1).

6) From the  we take also the following equations:

7) We shall transform the Equation (3) and take the Equation (4) and Equation(5) and Equation (6) into account:

8) So:

The Equation (8) can be found in  .

Figure 1. The Equation (8).

2.2. Derivation of the Golden Ratio

We mentioned in the introduction, the formula of golden ratio, we are going to derive it from a right-angled triangle  . (Note: Figure 2 can be found in  ).

9) See Figure 2:

10) The golden ratio:

11) See Equation (9):

12) Now we transform the Equation (10) so that it should be equal with Equation (8).We will take into account that:

13) Of course, the Equation (12) is identical with Equation (8):

Figure 2. In the book  the golden ratio = OE/OA, and.

3. Conclusion

It can be seen in the Equation (13) that equation can be expressed in more ways. But every equation is the equation of golden ratio. Naturally the golden ratio appears only then if

14) The Equation (13):

15) So:

4. Discussion

16) The Hardy equation  , and the Equation (13):

17) But still it is interesting the Equation (11) trivial solution:

Cite this paper
Jozsef, C. (2016) The Golden Ratio. Journal of Modern Physics, 7, 1944-1948. doi: 10.4236/jmp.2016.714171.
References
   Golden Ratio.
https://en.wikipedia.org/wiki/Golden_ratio

   Bronstejn I.N. and Szemengyajev, K.A. (1982) Matematikai zsebkonyv. Translate: Mathematical Pocketbook. Muszaki konyvkiado, Budapest.

   Einstein, A. (1921) Relativity: The Special and the General Theory. METHUEN and CO. LTD., London.
http://www.gutenberg.lib.md.us/3/6/1/1/36114/36114-pdf.pdf

   https://en.wikipedia.org/wiki/Lorentz_transformation

   Trasformazione di Lorentz.
https://it.wikipedia.org/wiki/Trasformazione_di_Lorentz

   El Naschie, M. (2016) Journal of Modern Physics, 7, 1420-1428.
http://dx.doi.org/10.4236/jmp.2016.712129

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