歮(Ji-u) 60stars Astrology Season 7 Mini-Appendix 4: A Brief Summary of Polygons and Circles～ " Φ-gon " and "１/Φ-gon "

60stars astrology

English version

By Tokyo-Tanuki

60stars Astrology Season 7

歮(Ji-u)  60stars Astrology Season 7 Mini-Appendix 4: A Brief Summary of Polygons and Circles～ " Φ-gon "  and  "１/Φ-gon "

1.  Now, I have been explaining the divisions (number of constellations) around polygons and circles at a rather fast pace so far, and some of you may be having a little trouble keeping up.

So, here is a brief summary.

The division of a polygon on a plane is,

Let "n" be the number of corners of an  " n-gon ",

Tn=3n(n+1).

...For example, for a hexagon,

T6 = 3 x 6 x (6 + 1) = 126,

A hexagon is made up of 126 constellations, or can be divided into 126 parts. That means.

These divisions are simply the sum of ,

the number of constellations needed for the edges, 

the number of constellations needed for the joints (clamps) of the edges, and 

the number of constellations needed for the reinforcements (diagonals) needed to maintain the shape.


For example, a square is,

Four sides,
Two diagonals
Four corners


The sides are made of hard bars, like iron, with a value of 6.

The diagonals (stiffeners) are also made of the same hard bar. Its value is 6.

There are two types of clamps, clamps on the outside and clamps on the inside, each of which has a value of 3.

If you put them on the inside and outside of a corner, you need two clamps = 6 constellations for that corner.

Then, for a square, the number of its constellations (divisions) is,

Side 4 x 6 = 24
Diagonal 2 x 6 = 12
Total of inside and outside clamps 6 x 4 = 24

So the total is 24 + 12 + 24 = 60, which means that 60 constellations are required for make a square.

T4 = 3 x 4 x (4 + 1) = 60, so it is as formulated.


2.  All polygonal divisions are indicated by this formula, but a few that are a bit confusing are 0-gon (0-gon), 1-gon (1-gon), 2-gon (2-gon), and 3-gon.

The 0-gon is a point.
Or more precisely, a position.
Therefore, it has no sides and, of course, no corners, so no clamps or diagonals.

T0 = 3 x 0 x (0 + 1) = 0
A 0-gon does not divide the perimeter.

A 1-gon is a straight line.
It has a starting point, but since it extends all the way to the end, it has no corners.
There are no clamps or diagonals.

T1 = 3 x 1 x (1 + 1) = 6
1-gon is divided into 6 constellations.

Straight lines are important because they make up the sides and diagonals of a multi-gon, which is itself a 1-gon, one of the polygons.



A 2-gon is a line segment, i.e., two straight lines between points AB, with a fixed mutual relationship.
It is a polygon, but because the angle is zero, it looks just like a straight line.

Since the angle is zero, there is no need for clamps on the inside of the corners, only on the outside.
There is no diagonal.

A 2-gon is,
T2 = 3 x 2 x (1 + 2) = 18
It can be divided into 18 constellations.

That's the total number of 2 straight lines and 2 outer clamps,

3.  .....Now, triangles (3-gon) .

Triangles are the most interesting because they are more primitive polygons than quadrilaterals or larger polygons.

The obvious difference is that the sum of the interior angles is only 180 degrees.

Then, as you would know if you have ever stood three pillars supporting each other, triangles, unlike quadrilaterals and larger polygons, do not need reinforcements (diagonals).

As a result, a triangle can be made with only 3 straight lines and 6 inner and outer clamps.

T3 = 3 x 3 x (3 + 1) = 36
The triangle is composed of 36 constellations (divisions).

The Keplerian triangle is very important as it shows the relationship between triangles, quadrilaterals and circles.

4.  ....This is the explanation of typical polygons.

And now we come to the special polygons described at the last part of this Season 7.

First, the phase in which two straight lines fuse rather than as inflection lines and become a curved line connecting a starting point and an ending point is called a "Φ-gon".

A straight line has only one direction, and two lines joined together have two directions.

This means that a Φ-gon can only connect two directions (90 degrees), and to bend it beyond 90 degrees, a new instruction (“add a curve”) is required.

A Φ-gon is actually one of the polygons with the characteristic of being a circle, although it was forgotten a long time ago.

Polygons become more complex as 1, 2, 3, .... and so on,
Curves increase in shape as 1.Φ, Φ², .... ... and so on.

.......Well, Φ and 1 are like twins or a married couple.




By the way, applying the formula, the number of constellations (divisions) that make up the Φ-gon is,

TΦ = 3 x Φ x (Φ + 1) = 3Φ³, 

so, it is made up of 3Φ³ constellations.


And if we subtract the total number of constellations in the original two lines (12) from the number of constellations in the 3Φ³ curve (Φ-gon),

3Φ³-12 = 0.708.... = 3/Φ³.

If we assume that this 3/Φ³ is  "1" , the number of constellations in the "Φ-gon" is,

3Φ³÷3/Φ³=Φ⁶.

Putting four 90-degree curves together, we can form the shape of a circle.

Therefore, the length of the circumference can be expressed as 4Φ⁶.
.....Let 1 be the apex angle (1/4Φ⁴) of the sacred right triangle with base angle 1/Φ and right angle part Φ²/4 used in the pyramid.

Then, when the circumference of 360 degrees is shown, the value is 71.777... = 4Φ⁶.


So the ancient Egyptians might use a uniform standard to measure the length and angle of circumference.



5.  Here it is important to note that even though the number of constellations (divisions) corresponding to the length of the circumference can be expressed as 3Φ³ x 4, it does not mean that a circle can be made with that number of constellations.

In other words, the number of constellations corresponding to the curve of the length of the circumference, and the number of constellations to make a circle are different.

In another words, four 90-degree curves can be arranged in the shape of a circle, but they are not a circle because they are not connected as they are.

Now, we need a clamp to connect the four curves, as explained in the previous article.

I will only write the conclusion here.
First, the number of a "clamp" is 3 (3 constellations).

What does 3 stand for? It is, well, dare I say it, "a line without direction".

In other words, it is what is called a point.

A 0-gon is also sometimes called a point, but to be precise, it indicates a position on a plane, and it is difficult to say whether it should be called a figure or not.

We use this "clamp" for circles as well as for polygons.

The value of the clamp is 3.
What is the angle of this?
Tn=3n(n+1),
The number of "n" for which Tn=3 is
"n" =1/Φ.

In other words, a clamp is a "1/Φ-gon".

When you ask whether this clamp can be used in the same way for a circle as it is for a polygon, the answer is that a slight deformation is required.

The reason is that a clamp is not a curve, so it cannot connect curves as it is.

A curve is the transformation of "two straight lines" numbers 12 into a "Φ-gon" number 3Φ³, which is a curve, so if we give the same transformation to the clamp, then,

3×(3Φ³/12)=3Φ³/4.

If we place this curved clamp in the gap between the four 90-degree curves, the circle is completed!

Then, we can conclude about the circle,
The circle can be made with (4 x 3Φ³) + (4 x 3Φ³/4) = 15Φ³ constellations.

To put it the other way around, since 3Φ³ is one "Φ-gon", a circle can be made with 5 "Φ-gon".

Therefore, to make a circle by connecting lines, we need 5 Φ-gons, or 10 1-gons (straight lines), right?

If the number of 1-gonal values is 6, we would need 10 of these, or the value 60, to make a circle.

It's  60stars!




6. Now, we have looked at things in a hurry so far.

The method of creating a circle described above is only for creating a circle by connecting lines.

This is not the same as creating a circle from a Golden Spiral.

As you can see by comparing the two, if you wanted to make a curve of the same length around a circle from the Golden Spiral, you would need six 90-degree curves.

In other words, there is an obvious difference. 

I will explain more about this when I have a chance.



7.  Finally, now is the start of summer.

This is the time of year when Sirius, the sun, and Canopus rise at the same time.

As I said last year, do not use magic during this time.
You will get caught up in strange things!




.......By the way, Φ and 1 are a straight line and a curve, or a couple, or twins, or, well, "a pair".

So, the number 1/Φ is considered sacred because it has the properties of 1 and Φ!

That's one of the reasons why this angle is used a lot in Egypt, ....well, just I guess.




NOW, season 7 is really over.

I have no plans for the future, but I won't be writing anything for a while!





Tanu-chan💓 TOKYO-TANUKI💛