淼 (hi-ro-i) 60stars astrology Mini Appendix off Season of season 6 "Who is Φ's friend after all?"

60stars astrology

English version

By Tokyo-Tanuki

60stars Astrology Off Season of season 6

淼 (hi-ro-i) 60stars astrology 

Mini Appendix off Season of season 6　
 "Who is Φ's friend after all?"

1.  By the way, as I have written, both (√7 + √3) /2 and √(√3 + 2) = (√6 + √2)  / 2 are cool, but they are not part of the so-called precious metal ratios (metallic ratios).

The metallic ratio is expressed by the equation

 {n + √(n² + 4)} / 2.

Φ (golden ratio) and τ (tau √2 + 1,  the white-silver ratio) , and (3 + √13) / 2, the bronze ratio, are metallic ratios. 

They are well known, respectively.

....But like (√7 + √3) / 2, which I mentioned earlier,

I think those ratios shown by ｛√(n+2)+√(n-2)｝/ √4  is also cool enough.

This ratio includes the golden ratio, the silver ratio, and the bronze ratio.

.... I'm not sure what happens when n < 2, though!

2.  So, as Tanu-chan wrote on Twitter,

｛√(n+2) + √(n-2) ｝/ √4 

(where “n” is usually an integer, maybe.) 

are cool, and may be friends of Φ.

These numbers are solutions that satisfy the equation,

X⁴-pX²+1=0 (where “p” is an integer).

....And especially the first five,

1, Φ, τ,

 

 (√7 + √3)/2,  

√(√3 + 2) = (√6 + √2) / 2 , 

and

(3 + √13) / 2  ( the bronze ratio which comes up a little later),

are very cool, I think!

The last one,  (3 + √13) / 2 = (√9 + √13) / √4, is not as fancy as Φ in the calculation, but perhaps has some relationship between the astrological formulas,

P = ｛(V x 2² / 3²） + √13)｝³

This solution will probably be important ...... Well, maybe!

3. .... By the way, about the above formula "X⁴-pX²+1=0",

"2”, which is the solution when p=17/4, is of course a cool number, but I don't think that "2" is cool in relation to this 🌀 equation " Q±(Xⁿ) ± R(1/Xⁿ) = P ".

Rather, I think it's cool because it's the only number where

 Q=R out of the numbers where P=QR=Q+R.

🌟　🌟　🌟　🌟

Well, so I think the first requirement for a friend of Φ is, 

"it has to look cool",

And second, "it gives a cool result when calculated", 

And, if possible, "to be related to the 🌀 equation" !

Let's look for cool numbers together!

....That's all for today.

Tanu-chan💓   TOKYO-TANUKI💛