The fractional parts of both Phi and its reciprocal (1/Phi) and its square (Phi^2) have the same fractional component.
Solving the quadratic equation 0 = x^2 – x – 1 produces the solutions:
Phi = (1+5^.5)/2 and Phi = (1-5^.5)/2
Taking the positive solution, you get Phi.
Phi = 1.61803398874989 1/Phi = 0.61803398874989 Phi^2 = 2. 61803398874989
However;
Solving the quadratic equation 0 = x^2 – x*i – 1 produces the solutions:
Phi(i) = (i+(i^2 + 1)^.5)/2 and Phi = (i-(i^2 + 1)^.5)/2
Taking the positive solution, you get Phi(i).
Phi(1) = 1.6180339887 1/ Phi(1) = 0.6180339887 Phi(1)^2-((1-1)/ Phi(1)) = 2.6180339887
Phi(2) = 2.4142135624 1/ Phi(2) = 0.4142135624 Phi(2)^2-((2-1)/ Phi(2)) = 5.4142135624
Phi(3) = 3.3027756377 1/ Phi(3) = 0.3027756377 Phi(3)^2-((3-1)/ Phi(3)) = 10.3027756377
Phi(4) = 4.2360679775 1/ Phi(4) = 0.2360679775 Phi(4)^2-((4-1)/ Phi(4)) = 17.2360679775
All positive integers produce a number whose fractional parts of both Phi(i), its
reciprocal (1/ Phi(i)) and Phi(i)^2-((i-1)/ Phi(i)) are the same.
Phi also can be used to create the “Golden Rectangle”. The golden triagonal is defined as a rectangle whose sides have the ratio of Phi. This rectangle has the property such that if a square is removed from the rectangle, a rectangle with the same proportion remains.
The other versions of Phi are similar. A rectangle whose sides have the ratio of Phi(i) will have the attribute such for every integer value of i, remove i squares and you are left with a rectangle of the same proportion as the original. In other words, for i = 1 where Phi(1) is 1.61803… remove one square, for i = 2 where Phi(2) is 2.414213… remove two squares.

If this has been documented elsewhere, please let me know where.

There’s an error in the text of the derivation.

The rearranged derivation is down to
B² – B – 1 = 0

Then the reader is introduced to a quadratic equation
ax² + bx + c = 0

But then this substitution is given as the formula for the golden ratio, which is now no longer a quadratic equation in one variable.
1a² – 1b¹ – 1c = 0

I assume what was meant was this:
aB² + bB + c = 0, where a=1, b=-1, and c=-1

1/1=1
2/1=2
3/2=1.5
5/3=1.666
8/5=1.6
13/8=1.625
21/13=1.61538

Can you explain the statement (PHI + PI) Tanh = 1

The series is actually 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597…………..
This is Fibonacci series.

For 2 3 8 21 55
#2+3(2)=8
#3+8(2)=19
so this is not a perfect sequence.

The sequence should include 5 13 34 for a complete list.

Thoughtonics says a super- verse got biggest crunch to be Shunyat ie. Zerot which is zero thought and that starts new. Super-verse.

I’m DDSharma, creator of physics of Thoughtonics, find golden ratio useful in my theory.