蟲(mu-shi) Season 6 Mini-Appendix 3 ～ Be a Friend of the Golden Ratio! Part 3

60stars astrology

English version

By Tokyo-Tanuki

60stars Astrology Season 6

蟲（mu-shi)  Season 6 Mini-Appendix 3

Be a Friend of the Golden Ratio!　Part 3

1.  So, we are looking for friends of the Golden Ratio (Φ),  but let's change our perspective for a moment!

In the case of ☈ = (1 + √13) / 2  = 2.302775637731... 

☈1 power =    ☈　

☈squared =    ☈ + 3

☈cubed =     4☈ + 3

☈4th power=  7☈ + 12

☈5th power =19☈ + 21

☈6th power =40☈ + 57

☈7th power =97☈ + 120

1+0, 1+3, 4+3, 7+12, 19+21, 40+57, 97+120, ...

Somehow, the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.... kind of atmosphere.

2.  Well, by the looks of it, and from the perspective of the coolness of the formula, let us examine the case of √41, i.e. the case of " (1 + √41) / 2 " , 

(1 + √41) / 2 = 3.701562118716....＝ ❁

Then,

 ❁ squared = 13.701562118716.... = ❁+10.

Also,

(❁-1)/10 = 1/❁ = 0.2701562118716....

1/(❁-1) = ❁/10 = 0.3701562118716....

If we multiply the powers in order, although the fractions don't add up ,but we get,

❁cubed = ❁ x ❁ squared = ❁(❁+10) = 11❁+10

❁4th power =❁(11❁+10) = 21❁+110

❁5th power = ❁(21❁+110) = 131❁+210

❁6th power = ❁(131❁+210) = 341❁+1310

1+0, 1+10, 11+10, 21+110, 131+210, 341+1310....

and it looks quite well-formed like the Fibonacci numbers !

The fractions don't line up very well, but maybe this is more interesting from a mathematical point of view!

3.  But, in order to be called a friend of "Φ" ,

(1) It looks nice, mathematically.

(2) The fractions of the multiplier are also aligned.

I think these two conditions are necessary!

So, let's look at the case of √37, i.e., the case of " (1+√37) /2 ",

(1 + √37)/2 = 3.541381265.... ＝If we set it to ✿,

Then, 

✿squared =  ✿ +9＝12.541381265.... 

✿cubed =  10✿ +9＝44.41381265.... 

What a surprise, the first, second, and third powers coincide as decimals (41381265....)!

Fancy, isn't it?

Simply put, 

✿ cubed - ✿ squared = 9✿, i.e., 9 = ✿squared - ✿= ✿(✿-1)

Then, further calculations,

✿4th power = 19✿ + 90

✿5th power = 109 ✿+171

✿6th power = 280 ✿+981

9/✿ = ✿-1

(✿-1) cubed = 10✿-19

.... in this case, the number "9"  (or total number  is "9" ) appears frequently for some reason.

That way, it doesn't look so bad!

Then this ✿ = (1 + √37)/2 = 3.541381265 .... could be a friend of Φ!  Maybe??

4. Tanu-chan has a reason to be so persistent in looking for Φ's friend.

I think this research might help us to the mysterious √13, which comes in the progression formula, i.e,

" P = { (V x 4 / 9) + √13 } cubed "

For this progression formula, we know the followings,

(1) It is related to the rotation of a rectangle.

(2) It is related to the actualization of data.

I explained a little about the 2:3:√13 right triangle when we talked about the King's Chamber in the Great Pyramid.

But Tanu-chan is still a little confused as to why this formula appears in astrology.

....If you have any ideas, please let me know!

🌟　🌟　🌟　🌟

By the way, come to think of it, there's a lot of research on Φ＝(1+√5) / 2, but,

(1+√3) / 2

(1+√7) / 2

I don't hear much research about them .... Let's get fantasizing again, shall we?

...So much for this series of "Searching for Φ's friends!".

Bye !

Have a happy New Year's Eve !

Tanu-chan💓   TOKYO-TANUKKO 💛