# Discrete Differential Geometry Assignment 0

# DDG Week2 Writing Assignment

## 2.1

Show that $V - E + F = 1$ for any polygonal disk.

For a simple n-sided polygon with n vertices, n edges, and 1 face, the equation above holds. When conncting another n-sided polygon to form a disk, the polygon can connect to the existing disk by merging $m$ edges. This will generate $n-m$ edges, $n-(m+1)$ vertices, and 1 new face.

$V' = V + n - m + 1 \\ E' = E + n - m \\ F' = F + 1 \\ V' - E' + F' = V - E + F + (n-(m+1)) - (n-m) + 1 = V - E + F = 1$

The last equality stems from our inductive assumption.

### 2.2

In a platonic solid, there are $F$ m, n-gons meeting at $V$ vertices.

$(F/m) - (nF/2) + F = 2$

### 2.8

Cl(S)

St(S)

Lk(S)

### 2.9

### 2.10

### 2.11

```
A0 = [
[1,1,0,0,0],
[1,0,1,0,0],
[1,0,0,1,0],
[1,0,0,0,1],
[0,1,0,0,1],
[0,1,1,0,0],
[0,0,1,1,0],
[0,0,0,1,1]
]
```

```
A1 = [
[1,0,0,1,1,0,0,0],
[1,1,0,0,0,1,0,0],
[0,1,1,0,0,0,1,0],
[0,0,1,1,0,0,0,1]
]
```

## Coding

Code is somewhere, I haven't decided where to put it. The screenshots below should verify that the c++ code is working to solve the exercises.

All tests green:

Star

Link

Closure