Golden Ratio

More coverage - external links :

Fibonacci Numbers, the Golden section and the Golden String by Ron Knott , The Golden Geometry of Solids or Phi in 3 dimensions

http://www.implosiongroup.com/ , https://www.theimploder.com/science/compressions-hydrogen-atom-and-phase-conjugation , https://www.google.com/search?q=golden+ratio+wave+nesting+implosion - much likely by Dan Winter (a "snail shell guy") and associates (← i cannot recommend nor not-recommend -- never had time to focus/verify ??? = read/"buy" with a precaution ...)

dd

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Compact/minimalistic - Random Facts : 

Naming differences on this site : k = φ = "Phi" = "Golden Ratio"

Decimal value : k ≈  1.6180339887498948482045868343656

Formula : ${\large \frac2{\sqrt{5}-1}=\frac{\sqrt{5}+1}2}$

Iterative : $k=1+\frac1k\quad,\quad k_{{}_{SEED}}=\frac{137·167}{020·707}=\frac{137·\left({13^2-2}\right)}{\left({13+7}\right)·\left({\left({3!}\right)!-13}\right)}={\small \left[437=137+300\right]}=\frac{\left({130+7}\right)·\left({437-270}\right)}{\left({13+7}\right)·\left({437+270}\right)}$

As a number system : k = x(D) where D=2 -- see below ↓↓

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From poly-equation : $k^2-k-1=0$

$$k=-\frac{-1}2±\sqrt{{\left({-\frac{-1}2}\right)}^2-\left({-1}\right)}=\cases{k\\ -\frac1k}$$

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Basic property 1 :

$$k^{n±1}=±k^{n}+k^{n∓1}\quad\left({\text{not forgetting that :: }k^{n±1}=k^{n}·k^{±1} }\right)$$

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Basic property 2 (applies to all series type of $a_{n+1}=a_n+a_{n-1}$) ::

$$a^{M(n+1)+n_0}=Look\left({M}\right)·a^{M(n)+n_0}-{\left({-1}\right)}^M·a^{M(n-1)+n_0}$$

where M , n , n₀ are ±Integers ,
a is ±Realletter ,
$Look(M)=k^M+(-1)^M·k^{-M}$ is the M-th indexed member of the Lucas-series

$$3_{\ -2}\ ,\ -1_{\ -1}\ ,\ 2_{\ 0}\ ,\ 1_{\ 1}\ ,\ 3_{\ 2}\ ,\ 4_{\ 3}\ ,\ 7_{\ 4}\ ,\ 11_{\ 5}\ ,\ 18_{\ 6}\ ,\ 29_{\ 7}$$

example ::

$$\cases{k^{25}=11·k^{20}+k^{15}\ //\ k^{5·1+20\\ 5·5}=11·k^{5·0+20\\ 5·4}+k^{5·(-1)+20\\ 5·3}\\{\ }\\ k^{36}=18·k^{30}-k^{24}\ //\ \cases{17.94427191·k^{30}=18·1·k^{30}-0.05572809·k^{30}\\33385282=33488964-103682}\\{\ }\\ k^{69}=29·k^{62}+k^{55}\ //\ k^{70-1\\ 49+20}=29·k^{63-1\\ 42+20}+k^{56-1\\ 35+20}}$$

Def. :

$\displaystyle{Fibo(K)=\frac{k^M-(-1)^M·k^{-M}}{\sqrt{5}}}$ is the K-th indexed member of the Fibonacci-series

$$-1_{\ -2}\ ,\ 1_{\ -1}\ ,\ 0_{\ 0}\ ,\ 1_{\ 1}\ ,\ 1_{\ 2}\ ,\ 2_{\ 3}\ ,\ 3_{\ 4}\ ,\ 5_{\ 5}\ ,\ 8_{\ 6}\ ,\ 13_{\ 7}$$

example.2 ::

$$\cases{Fibo(10)=Look(3)·Fibo(7)+Fibo(4)\ //\ 55=4·13+3\\{\ }\\Look(N)=Look(0)·Look(N)-Look(N)=(2-1)·Look(N)}$$

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Functions D(x) , x(D) ::

Def. & properties :

$$x(D)·x(\frac1D)=x(D)+x(\frac1D)\ \Rightarrow\ \frac1{x(D)}+\frac1{x(\frac1D)}=1\ \Rightarrow\ x(D)-1=\frac1{x(\frac1D)-1}$$

$$x(\frac1D)=x(D)^D\ ,\ x(D)=x(\frac1D)^{\frac1D}\ ,\ x(D)^\sqrt{D}=x(\frac1D)^\sqrt{\frac1D}$$

$$Def.\ :\ D(x)=1-\frac{ln\left({x-1}\right)}{ln\ x}=\frac{ln\left({1-\frac1x}\right)}{ln\ \frac1x}\\ D(x)=1-log_x\left({x-1}\right)=log_{\frac1x}\left({1-\frac1x}\right)$$

$$Def.\ :\ x^D-x^{D-1}-1=0\ \Rightarrow\ x^{D-1}=\frac{x^D}x=\frac{x^D-2}{2-x}=\frac1{x-1}$$

D = 1	x = 2	12 = 0.1(1)2 = 0.11111111...2
D = 2	x = k ≈ 1.618	10k = 1.1k = 1.01(01)k = 0.1(1)k = 0.11111111...k
D = 3	x = t ≈ 1.466	100t = 10.1t = 10.01(001)t = 1.011(011)t =    .101(101)t +   0.010(010)t   = 0.1(1)t = 0.11111111...t

the above table derives from : $\displaystyle{\mathbf{x^{D-1}=x^D-1}=\frac x{x-1}=\mathbf{\frac1{1-\frac1x}}}$ 

about Geometric series (← notice the r⁰ at the beginning of the S = 1 + r + r²... so , $S-1=...=x^{D-1}$)

 

x(D) graph. ( x - vert. , D - horiz. ) :

 

note : The x is a number system base that has a FP base !!!

 

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