驫（hyo-u) 60 Stars Astrology Season Holiday " An Extra 6" Tanu-chan's Concerns Part 3 “71.777.”～What is the relationship between this number and the golden spiral?

60 STARS ASTROLOGY
ENGLISH VERSION
SEASON HOLIDAYS "An Extra"

 

驫（hyo-u) 60 Stars Astrology Season Holiday " An Extra 6 "　

Tanu-chan's Concerns Part 3   “71.777.”～What is the relationship between this number and the golden spiral?

1.  Right-angled triangles are often used in pyramids, where the base angle is 1/Φ and the right-angled part is Φ²/4,and the vertex angle is 1/(4Φ⁴).

By the way, as I wrote in my Twitter post, if the angle of the vertex angle(1/Φ）＝ 1, 

360 degrees (360°)  is expressed as 71.777.... = 4Φ⁶


On the other hand, also I wrote this at the end of Season 7, the length of the circumference is the sum of 4 Φ-gon, so the length is 4 x 3Φ³ = 12Φ³ (i.e.,Φ-gon is 3Φ³）

This can also be expressed as 4Φ⁶, by multiplying Φ³/3.


So, the pyramid people were able to tell the length of the arc by the angle, in the same time.

Well, it is an ancient version of the radian concept in mathematics.


But it is still amazing because the Egyptians thought about what a circle is, and from there, they theoretically aligned the units of a circle and an angle.

2.  By the way, 4Φ⁶=71.777.... and this is,

71.777 = 32Φ + 20
 　　　= 20Φ² + 12Φ
 　　　= 16Φ³ + 4 = 12Φ³ + 8Φ²
　　　 =10Φ⁴ + 2Φ = 8Φ⁴ + 4Φ³
　　　 = 6Φ⁵ + 2Φ² = 4Φ⁵ + 4Φ⁴
　　　 = 4Φ⁶ + 0
　　　 = 2Φ⁷ + 2Φ⁴

and so on.....from Φ to Φ⁷, which can be expressed as the sum of two terms.

Well, they all, when calculated, are 71.777... so, it can be expressed in a variety of ways.





3.  Now, just to change the subject a bit, I already wrote about this in my Twitter post,

Regarding the relationship between a circle and a kind of logarithmic spiral (GOLDEN SPIRAL, hereinafter referred to "spiral" ) as shown in the figure below, if we align the lengths of the 90 degree arcs of the last part of the spiral and the circle, the ratio of the overall lengths is,

spiral : circle = Φ² : 4.

It may be strange. We think the spiral should go on forever....
(Fig.1)

Anyway, then, at this stage, Φ² = 2.618033988... so circle (4) is definitely larger (longer circumference), 

...and at this stage, if we assume that the length of the entire circumference is 4Φ⁶ instead of 4, the length of the line of the spiral would be Φ⁸.

(Fig.2)

4Φ⁶ = 71.777...
Φ⁸ = 46.978713......
Since the circumference is longer, let's advance the spiral another 90 degrees.
(Fig.3)

Φ⁸ + Φ⁷ = 76.0131556... 
So the length of the spiral would be 76.0131556....


Then the total length of the spiral would be longer than the length of the circumference.

So to make the lengths equal,
76.0131556..... to 71.777.... minus 4.2360679...... = Φ³ is to be subtracted from the length of the spiral.

....Then, Φ⁸ + Φ⁷ - Φ³ = 71.777.... and the lengths of the spiral and the circle are equal.

4.   Now, if we look closely at the equation Φ⁸ + Φ⁷-Φ³ = 71.777.... we realize,  "Φ⁸ + Φ⁷" is Φ⁹.

So we can transform it into Φ⁹-Φ³＝71.777....

Then, as before, the spiral will indicate the length of the circle by the coefficient of two terms.

...Is the spiral a member of a circle?　


On the other hand, the coefficients are negative (Φ⁹-Φ³＝71.777....,) which is different from the case where the coefficients are positive, such as "32Φ + 20" or "20Φ² + 12Φ", which I wrote about earlier.

What exactly is a spiral?

This is how the mystery deepens.

Is a Golden Spiral a member of a circle?　

Or is it a member of a line like a Φ - gon or a 1 - gon?



5.  Well, that's why ...... this is the end of the day, as in the previous issue, with nothing resolved.

..............Anyway, it  almost comes the autumn! 

It's time to wrap up this Extras ("Season Holidays") here, and get to the next season.

Hey, it's "Season Vacation" !

Maybe starting in October ? maybe !



Tanu-chan💓 TOKYO-TANUKI💛