磊 (sa-za-re-i-shi) Season 6 Mini-Appendix 3 ～ Let's find friends of the Golden Ratio!

60stars astrology

English version

By Tokyo-Tanuki

60stars Astrology Season 6

磊 Sa-za-re-i-shi　Season 6 mini-appendix 3　

Part 1: Let's find friends of the Golden Ratio!　

1 .Well, Let us take a short break from learning horoscopes.

Now, I'm running out of Greek characters to use for the mini-appendix, so from now on I'll use Kanji characters (Chinese characters) .

I'll use kanji that have three identical parts, like

"森"（"Mo-ri" -forest） 

and 

"州" ("Ku-ni" -state）, etc.

Today's is "磊"' ( sazare-ishi ) , it  means "a lot of small rocks" in Japanese. 




2.  So far, I have sometimes talked about the Golden Ratio Φ = (√5 + 1)/2 = 1.61803988.... 


we have been talking about the following.
Φ = 1.618033988
1/Φ = 0.618033988
1 + Φ = Φ squared = 2.618033988

"Fraction of 18033988...." 

Tanu-chan would like to have a friend for "Φ", whose fractions are mysteriously aligned, somehow similar type.

Well, in the case of Φ, it's flashy because the fractions are similar "18033988...." when it is -1th,＋1th,and 2th powered.

And Tanu-chan thought, “It need not to be this flashy, but is there any other baby that looks like Φ ?"

So, I decided to search for Φ's friends, at first, look like the same "squared" or "cubed" type.  


What I am about to tell you may have a little something to do with the Golden Ratio or the Number of Lucas.


🌟　🌟　🌟　🌟

3. This question is more difficult than it seems, so I went back to the basics.

First, "Φ" is the solution to  "X squared - X - 1 = 0",  right?

Well, to put it simply, if you look for "X"  such that  "X+1" becomes "X squared", then the answer is Φ.

So, now I look for a positive integer such that 

" X + a "= "X squared " .

From the quadratic formula, we find the following,

X = {1 + √(1 + 4a) } / 2

....And,

First, substituting "a = 1", we get X = Φ.

Next, if "a = 2", then X = (1 + √9)/2 = 2.

Well, "2" is a very special number, 

2＋2 = 4 
2×2 = 4.

But we all know that  "2"  is  "very special".
So to speak, "2"  is a famous idol, not a friend.

..."Φ" is famous among maniacs, but in general, it is not as famous as "2".



4.  So let's substitute "a = 3"

Then we get X = (1 + √13)/2.

If we calculate it,

X = (1 + √13)/2 = 2.302775637731....　

and

X squared = 5.302775637731...

So, we have the same fraction for now.

....Well, to tell the truth, it is always the case with this quadratic formula. They have the same fractions when squared.


However, this alone does not explain the uniqueness of this number  " (1 + √13)/2" .

If we cube this, we get
12.211102550....

This sequence of numbers "11102550...." is a bit interesting.

Since it is too long to write down every fractional number, we use the weather symbol for lightning, “☈”
　
☈ = (1 + √13)/2 = 2.302775637731... Let's say!

Because “13” and “☈” are a bit similar in form.

Then, 
☈ cubed = {(1+√13)/2} cubed = 12.211102550...

Furthermore, the square of the figure is
☈ 6th power = {(1 + √13)/2} 6th power = 149.11102550...

.....the fractions of the result are almost perfectly equal.


And, in addition, 
☈ x 4 = 9.211102550....

Therefore, we have

☈×10=23.211102550....
☈×40=92.211102550....

They all end in "11102550....."

Also, ☈×√208 = 33.211102550....

Oh!  Wow, what's this?　What's new to discover?

....Well, originally,

 (√13 ) / 5 = 0.7211102550...., and, 2×√13=7.211102550....

So, the fraction "11102550.... " is appearing is not strange in itself.

But when so many of them appear together, it seems that ☈ has some special property, similar to that of "Φ" !

☈ Cubed = (☈ x 4) + 3,  

which seems kind of cool！




5. ....By the way, to tell the truth, the above

☈ Cubed = (☈ x 4) + 3

is an actually common style in the form 

" S = {1 + √(1 + 4a)} / 2 " .


For example, when a = 1, that is, when S = Φ,
S cubed = Φ cubed = (Φ × 2) + 1 

= (S × 2) + 1.

Next, for example, when a = 4, S = (1 + √17) / 2,
S cubed = {(1 + √17) / 2} cubed = [{(1 + √17) / 2}] × 5 + 4 

= (S × 5) + 4

And further, when a = 5, S = (1 + √21) / 2,
S cubed = {(1 + √21) / 2} cubed = [{(1 + √17) / 2}] × 6 + 5 

= (S × 6) + 5


...That is, when a = X (X is a positive integer), and S = P = {1 + √ (1 + 4X) / 2 ｝, in general,

" P Cubed = (P x Q) + R = {P x (X + 1)} + X "　



Well, so it seems that there are quite a few numbers that could be more than Φ's acquaintance.

Some of you may say, "What the heck, I didn't know that !! "



6.  .... However,
When a = 2,  i.e.,
When S = ☈ = (1 + √13) / 2,  things are a little different than usual.

That is, as I mentioned earlier, in the case of ☈,

☈Cubed = (☈ x 4) + 3 = 12.211102550...
☈6-power = (☈×40) + 57 = 149.11102550...,

the fractions of the two powers (cubed and 6th power) come together.
This phenomenon is not seen when a = 1 (S = Φ).

☈ = (1+√13)/2 = 2.302775637731...
☈×4= 9.211102550....　
☈ Squared = 5.302775637731.... = ☈+3
☈ Cubed = (☈×4) + 3 = 4☈ + 3 = 12.211102550...　
☈ 6th powered = (☈×40) + 57 = 149.11102550...


Hmmm. Fraction “302775637731...” and “11102550...” seem to be appearing alternately, don't they?



This will be continued next time!




Tanu-chan💓 TOKYO-TANUKI💛