I saw a new market at a crazy party in Akihabara

The other day, I went to a year-end party in Akihabara. It was really exciting and crazy.

The party was thrown by 3D-GAN (3D-Geometry Application Network), which is an NPO and was established 2 years ago in Akihabara. 3D-GAN's purpose is to share and promote 3D shape data/technology more. Various people were there from various industries.

Look at the above picture. The guy wearing the strange mask is me. The mask is about a character named Mona, which is popular among Internet users in Japan.

　　 ∧＿∧　　／￣￣￣￣￣
　　（　´Д｀）＜　オマエモナー
　　（　　 　 ） 　＼＿＿＿＿＿
　　│ │ │
　　（＿_）＿）

I borrowed the mask from a fellow geek attending the party, who had made the mask. It was really surprising that the mask had a tiny CCD camera on the face and had a LCD(liquid crystal display) inside, so I wouldn't be blind while wearing it.

Akihabara is the only city where we can meet such interesting geeks. Yes, you know, Akihabara is a city for various kinds of geeks, nerds, maniacs, and Otaku.

The people in the party were diverse. They were of various ages, from various industries, and had various interests. The only common interest among them was 3D technology.

They would never have met if it weren't for 3D data.
They would never have met if it weren't for 3D-GAN.
But they met. The encounter was exciting. I believe the encounter will drive a change. The change is about the spread of 3D data and technology.

I remember Mr. Soma, a founder of 3D-GAN, told me that establishing 3D-GAN in Akihabara was significant. He said that there were no cities in the world more unique than Akihabara, and he believed an innovation would occur from a unique city like Akihabara. He chose Akihabara strategically.

By the way, I found a Hisui user at the party(Hisui is a CAD framework I developed). He told me that he was using Hisui to convert obj format file to STL format file. It made me happy to talk to a user of my own product.

He told me that he couldn't find any software which can convert obj format to STL format except Hisui. I was surprised because to develop the function converting obj to STL was really easy. But I know that obj format is used by CG modelers and STL format is used by manufacturers, and there is a deep gulf between them. It seems that they don't exchange any information with each other. So, I guess no vendors have found the need of the conversion. But I could see it in Akihabara.

So, let's meet in Akihabara.

Euler's Formula in Topology

CAD is based on mathematics. The formula I was impressed by the most when I joined the CAD industry was Euler's formula.

v - e + f = 2(1 - g)

Where 'v' is the number of vertices, 'e' is the number of edges, and 'f' is the number of faces. 'g' means 'genus', which I will explain later.

I was surprised by the fact that every polyhedral object satisfies this formula. In this entry, we will verify this formula with some concrete examples. Let's begin with a simple cube.

First, you should keep in mind that the genus of a cube is zero. You will know why later. Then, we can simplify the formula as below.

v - e + f = 2

What we should do is to count vertices, edges, and faces in a cube. It's really easy work. The 'v'(the number of vertices in a cube) is 8. The 'e'(the number of edges in a cube) is 12, and the 'f'(the number of faces in a cube) is 6. Substitute these numbers in the formula.

v - e + f = 8 - 12 + 6 = 2

Bingo! This simple cube satisfies Euler's formula indeed.
Next, you might wonder what will happen if each quadrangle face of the cube is split into 2 triangles. Let's verify this case.

In this case, e becomes 18(= 12 + 6) and f becomes 12(= 2 * 6) because each face is split into two triangles with one diagonal edge. v remains 8, of course. Substitute these numbers in the formula.

v - e + f = 8 - 18 + 12 = 2

Bingo! This is amazing, isn't it?

Next example is a little complex. Can you count v, e, and f in this object?

v is 16, e is 32, and f is 16. Let's substitute these numbers.

v - e + f = 16 - 32 + 16 = 0

Oops! What is wrong?
Now, you need to know what the 'genus' is. The genus means the number of holes in an object. Because there is one hole in the above object like a doughnut, the genus 'g' is 1. Now you can understand why the genus of a cube is zero. So, In this case, Euler's formula must be written as below.

v - e + f = 0

So, the doughnut satisfies Euler's formula indeed.

This theory is categorized as topology. Note that geometrical shapes don't matter in topology. Only linkages between vertices, edges, and faces matter. Topology interested me because it's really different from geometry I learned at high school.

Every solid kernel, which is a calculation engine of 3D CAD, is equipped with 2 kinds of operations. The one is a kind of geometrical operations, and the other is a kind of topological operations, which is called Euler operations. I feel it's amazing that such abstract mathematics is alive in CAD software. Don't you think so?