㉟ Circle and Sphere～7 Aspects and 7 Regular Polyhedra 60stars astrology (season3)

60stars astrology Season 3

By Tokyo-Tanuki

English version

㉟ 7 Aspects and 7 Regular Polyhedra

1. In the previous article, I wrote about the relations between the 5 regular polyhedra and the aspects.

But, If you like geometry, you might think

"Tanu-chan can explain the 120-degree trine △ or the 90-degree square🔲 and so on, but have not explained the 0 degree conjunction☌, and 180-degree Opposition☍! "

"There are only 5 regular polyhedra, math proves it."

"Are you crazy, Tanu-chan?"

2. And that's what I'm going to explain this time. 

Tanu-chan is a liberal arts student. I'll try to make it simple.

Everyone thinks that there are 5 regular polyhedra. 

There are the following assumptions.

"The faces of polyhedra must be able to be written as regular polygons."

For example, the faces of a regular hexahedron are made of squares, and the faces of a regular dodecahedron are made of pentagons.

In Tanu-chan's view, it is not assumed that the faces of a regular polyhedron are made of regular polygons.

What Tanu-chan means is that a regular polyhedron is a three-dimensional object that has the same correct shape (regular polygon or circle) for all of its faces in a three-dimensional space.

3.Okay, I'm going on explaining it.

It's called Euler's Polyhedron Theorem (Euler's formula).

Number of vertices V - Number of edges (edges) E + Number of faces (faces) F = 2　

In short, we have,

V-E + F = 2

In other words

F (number of faces) = 2-V + E 

This is the formula for a polyhedron.

So first, let's consider a plane. 

A plane in three-dimensional space has a back and a front. So it is not a monohedron, but a dihedron.

Now, if we consider a plane regular circle 〇 (disk), the back and front have the same shape (circle).

The vertex can be anywhere on the regular circle, but if we take a vertex anywhere on the regular circle, the number of edges is only one, since there is only one line segment (circumference) that goes around the circle and back to the origin.

And the vertex can be anywhere on the circumference, so the number of vertices V is infinite (∞), the number of edges E is the same as the number of vertices, so E is infinite (∞), and there are two faces (F). 

Then, if we force this into Euler's formula that we just wrote, we get,

V-E + F = ∞ - ∞ + 2 (since it is a dihedron) = 2　

To put it more simply.

F = 2-∞ + ∞ = 2

which means that a regular dihedron is a regular circle(disk).

In other words, a regular dihedron is an equilateral circle (in three dimensions).

Let us view this regular dihedron from its vertices in three-dimensional space, as we did with Plato's regular polyhedron.

Then,OH!　　　　　　　　　　　　　　　　　　　　　　　　　"━" A straight line appears.

So, in this case, the straight line viewed from the vertex of the regular circle divides the ecliptic circle, so the angle is 180 degrees.

This explains "the opposition☍".

Are you OK?　

If you're not interested, just skip it.

3. Next, I'll go into the explanation of "the conjunction☌"

I'm sure those of you who have already figured it out understand it.

First, consider a sphere in three dimensions.

If a vertex is placed somewhere on the surface of the sphere, the number of edges is infinite (E＝∞), assuming that the edges rotate in the same way as in a circle.

Since the number of vertices is also infinite (V

＝∞), the number of edges is the square of infinity.

And

Euler's formula, V-E + F = 2, is transformed to 

F = 2-V + E.

And then, by applying this formula to "V=∞" and "E=∞ squared", respectively, we get

F = 2-∞ + square of ∞.

To put this in order, we have

F = ∞ (∞-1) + 2　

This may seem a little strange, but it means that the sphere is 

∞ (∞-1) + 2　polyhedron.

And If we look at the sphere from its vertex and apply it to the ecliptic circle, it overlaps the ecliptic circle and does not divide the ecliptic circle at all. 

In other words, its aspect is 360 degrees, or 0 degrees (conjunction ☌).

4. In other words, the angle at which the seven regular polyhedra (icosahedron, tetrahedron, hexahedron, octahedron, dodecahedron, dihedron, and sphere) divide the ecliptic circle is called "the aspect".

Of course you can ignore them because they are complicated. 

Tanu-chan is just a liberal arts student　((´∀｀*))

5. Here is an additional explanation for those who are just a little bit interested.

Look at the equation of the sphere I just showed you.

F = ∞ (∞ - 1) + 2

Let's replace infinity ∞ in parentheses with "1" in the Formula for Sphere.

Then, we get

F = ∞ × (1-1) + 2 = 2-∞ + ∞ = 2.

This is exactly the same formula as the dihedron (F = 2-∞ + ∞ = 2).

So the dihedron (regular circle) and the sphere are in the same family. 

So ∞ circles come together to form a sphere!

6. To sum up the above in a nutshell, the number of regular polyhedra is 7.

Polyhedra  1 ⇒ type that evolves into a sphere

Polyhedra 2 ⇒ type that increases the number of faces of regular polygons

So there are these two types of polyhedra.

And there are five regular polyhedra between the heavens (spheres) and the earth (planes). 

This is the area of human activity.

So the number representing the human being is 5.

When heaven, earth, and man are combined, it becomes the whole world, and the number is 7.

Heaven is to save people who are attracted to hell (5 + 1 earth = 6).

Yes, thus, "7" means to ward off evil.

7.This has been known for a long time. 

You know, there are groups with symbols that combine a ruler (polygon) and a compass (circle).

Also You know, there are forward-rear round burial mounds in Japan.

And that's what those means.

It is not a coincidence that the picture of an ancient Egyptian ship and the picture of an ancient ship from a Japanese burial mound are the same.

That was a bit tedious today, wasn't it?

I'll make it easier next time!

Tanu-chan🔓 TOKYO-TANUKI💛