The classic example of a two-dimensional homology class is the "void" surrounded by a sphere in three-dimensional space.
Challenge question: **Can two-dimensional topological voids arise from point clouds in two-dimensional space?**
We will answer this question programmatically by computing Vietorisâ€“Rips persistent homology of random point clouds in the square $[0, 1] \times [0, 1] \subset \mathbb{R}^2$.

If you are looking at a static version of this notebook and would like to run its contents, head over to GitHub and download the source.

**License: AGPLv3**

In [ ]:

```
import numpy as np
from gtda.homology import VietorisRipsPersistence
import itertools
import matplotlib.pyplot as plt # Not a requirement of giotto-tda, but is needed here
np.random.seed(1) # Set numpy's random seed
```

In [ ]:

```
# Initialize the Vietorisâ€“Rips transformer
VR = VietorisRipsPersistence(homology_dimensions=(2,), max_edge_length=np.inf)
```

In [ ]:

```
# Create n_samples point clouds of n_points points
n_samples = 15000
n_points = 6
point_clouds = np.random.random((n_samples, n_points, 2))
# Compute persistence diagrams of all point clouds
diags = VR.fit_transform(point_clouds)
```

In [ ]:

```
diffs = np.nan_to_num(diags[:, :, 1] - diags[:, :, 0]) # Compute lifetimes
indices = np.argwhere(diffs != 0) # Indices with non-zero lifetime
print(f'There are {len(indices[:, 0])} persistent homology classes in dimension 2 across all samples.')
print(f'There are {len(np.unique(indices[:, 0]))} different point clouds with at least one persistent homology class in dimension 2.')
```

We can now plot the edges which exist when these persistent homology classes are born. What do the clique complexes of the resulting graphs remind you of?

In [ ]:

```
for i in indices[:, 0]:
for e in itertools.combinations(point_clouds[i], 2):
if np.linalg.norm(e[0] - e[1]) < diags[i, 0, 1] - 0.00001:
edge = np.stack([e[0], e[1]])
plt.plot(edge[:, 0], edge[:, 1])
plt.show()
```