Making the World's First Kusudama with AI Part 10: Unit β-γ

Hello, good morning, good afternoon, or good evening. Can you tell what the thumbnail is showing?

A kusudama modeled after a meteorite

Let's look at this diagram first.

As the title suggests, I use Unit β and Unit γ.

What are these for? Take a close look.

Unit β is for connecting Unit B and C.
Unit γ is for connecting Unit A and C.

Appearance

Even among my kusudama, I made this without using much brainpower. Now, let's look at the real thing.

Well, if you call it a meteorite, is it a meteorite...?

Gradation was also something I was a bit particular about.

Purple → Blue → Slightly lighter blue.

When I searched for images of meteorites, only red ones came up.
A bit of a regret there 💦.

Mathematical structure

I think there are some parts that might be hard to understand.
Feel free to skip them.

You can find the details in Part 0, but I will explain the structure of this kusudama.

This is a shape where eight triangular pyramids are attached to a regular octahedron base.

This time, there are a total of three types of triangular pyramid shapes.

Each is a regular triangular pyramid where the apex angle of the isosceles triangle on the side is 90°, 60°, and 45°.

And there are four, three, and one of each, respectively.

By the way, there is also a kusudama with an icosahedron base, which you can see in Whimsical Kusudama ③.

Number of units required

In this case

I will introduce the units required this time.

Normally, the number of units required is,

This time, 8×3/2=12. We are using a total of 12 units, right?

The reason for this is that when you attach a polygonal pyramid to the base, the number of faces that exist there isthe number of edges that one face of the base has.

And,that face is proportional to the number of faces the base has.

One unit creates two faces.

From this, that formula is derived.

Other cases

There are unpleasant exceptions.Exceptions are part of mathematics, aren't they(?).

What if it's a shape where polygons are not attached?

In other words, the case where you make the base polyhedrondirectly.

In that case, it's not impossible to do it with the conventional method, but it is
overwhelminglyinefficient.

Unit A aside,there is a way to force it using Unit C.

I cannot explain it this time, so please refer to ④ through ⑥.

That was all(?).