176 60 Stars Astrology Season 7 Appendix 7: Basis for how to determine the Number of Constellations(Divisions) around polygons..?

60stars astrology

English version

By Tokyo-Tanuki

60stars Astrology Season 7

60 Stars Astrology Season 7  Appendix 7: 
Basis for how to determine the Number of Constellations(Divisions) around polygons..?

1.  Now, I feel like I'll be taken to the hospital very soon, so I'm going to continue this appendix with the previous one, but I'll do it quickly.

Today I am going to talk about polygons, not circles.


The story of circles is a bit more complicated and would be longer, so I tried to write about it before, but failed.

I have to finish Season 7 by in this May or June, at most the summer solstice, so I might not make it in time.

....So maybe someday in the future, I will write about Circles....if possible. 


2.  Now, the number of constellations around the polygon, i.e., the number of polygonal segments （divisions）,

"As the number of angles = n,

The numbers of the constellations of the figure is 

   

 T(n) = 3n(n+1)."


For example, if it is a triangle, the number of angls are three,

 

So the numbers of the constellation is,

Since n = 3, T(n = 3) = 3 x 3 x (3 + 1) = 36 (segments).



Also, if it is a heptagon, the number of segments of constellations is,

Since n = 7, T (n = 7) = 3 x 7 x (7 + 1) = 21 x 8 = 168 (segments).

Today, I will explain why this is so.

3.  Before that, what I want you to check first is,

A. This blog is made up of Tanu-chan's delusions.
B. Tanu-chan is a humanities student.
C. All of this is just an image.



...So, there is no need to listen to me seriously.



.....OK, now for a brief explanation.

First, polygons are made of a different material than circles.
As an image, it can be a steel rod or something hard.

And, for example, a triangle is,
First, you need three hard iron bars.
Each bar is a straight line (a 1-gon), so it contains 6 constellations.

At first, we need 6 x 3 = 18 constellations.

However, the three bars do not stick together and stabilize on their own.

Therefore, clamps (fixing tools) are attached to the inside and outside of the corner to stabilize it.

The number of constellations contained in a clamp is 3 constellations per clamp


Since there are three corners, 6 x 3 clamps are needed, for a total of 18 constellations.


So, The entire triangle needs, 
18 constellations (straight lines) + 18 constellations (clamps) = 36 constellations

The total number of constellations in the triangle is 36.




4.  Explain it this way, for example, since a square has four sides and four corners,

you need at least,  
4 iron bars
8 clamps (4 places x 2 inside and 2 outside = 8 clamps)
Because it is necessary,　

Then, maybe you think,
 
(4 x 6 constellations) + (8 x 3 constellations) = 48
Will there be a total of 48 constellations of a square?


.....but that is not correct.

If they are not open beyond 90 degrees like triangles, they(the bars) support each other and do not need supporting rods such as diagonals.

When you put several rods on the ground, they will stand if they support each other, just like that.

Clamps are necessary, though.


However, if the angle of the corners exceeds 90 degrees, as in a square, the shape is not stable as it is. 

In other words, you need to put in a diagonal brace to stabilize the bar, or put in a reinforcing material with the same strength as the brace at the corner.

Well, to put it simply, you need diagonals.


And those diagonals and reinforcements are also bars (made) of steel, so the number of constellations per diagonal is 6.


In the case of a square, two diagonals, or two reinforcing pieces, go in.

...So, finally,
48 constellations + 2 reinforcers (6 x 2) = 60 constellations
is the total number of constellations.

🌟　🌟　🌟　🌟

Then, for one more example, in the case of an octagon,

8 steel rods,
8 clamps x (3 inside + 3 outside)
20 reinforcements (diagonals)
is required.

So the total number of constellations is,
(8 iron bars (straight lines) x 6 constellations) + (8 clamps x (3 constellations + 3 constellations)) + (20 reinforcements or diagonals x 6 constellations) = 216 constellations

Thus, we need 216 constellations.


Now, let us apply the formula T(n),
T (n = 8) = 3n (n + 1) = 3 x 8 x (8 + 1) = 216

So, it fits perfectly!



It's a bit tedious, but also for one more example, in the case of an 11-gon,

There are 11 bars,
11 clamps x (3 inside + 3 outside)
44 reinforcements (diagonals)

Therefore, the total number of constellations is,
11 bars x 6 constellations = 66 constellations
Clamps 11 places x (3 constellations + 3 constellations) = 66 constellations

Reinforcement 44 pieces x 6 constellations = 264 constellations

for a total of 396 constellations.

Applying the formula,
T (n = 11) = 3 x 11 x (11 + 1) = 33 x 12 = 396

This also fits.



5.   By the way, in the case of a 2-gon, since the lines overlap and the inside cannot be seen, there is no need for a reinforcement or even an inside clamp, only an outside clamp is needed to prevent slippage!

i.e.,

2 straight lines, so 6 constellations x 2 = 12 constellations
Since there are only 2 clamps on the outside, 2 x 3 clamps = 6 constellations

Total 18 constellations.


Of course, in the formula,
T (n = 2) = 3 x 2 x (2 + 1) = 18

and the answer is correct.

....Well, I guess it goes like this. 

Do you understand "the equation for polygons ( T(n) ) ?"


And I will explain about circles in the future... maybe... if possible.

Of course, all of the above is just my imagination!

That's all for today.





Tanu-chan💓 TOKYO-TANUKI💛