#!/usr/bin/env python
# coding: utf-8

# # Can two-dimensional topological voids exist in two dimensions?
# 
# #### License: Apache 2.0

# 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 Vietoris-Rips persistence homology of random point clouds in the square $[0, 1] \times [0, 1] \subset \mathbb{R}^2$.

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import numpy as np
from giotto.homology import VietorisRipsPersistence as VR
import itertools

import matplotlib.pyplot as plt


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# Initializing the Vietoris-Rips transformer
vr = VR(homology_dimensions=(2,), max_edge_length=np.inf)
n_samples = 15000
n_points = 6


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# Create n_samples point clouds of n_points points
PCS = np.random.random((n_samples, n_points, 2))  
# Compute persistence diagrams of all point clouds
DGMS = vr.fit_transform(PCS)  


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diffs = np.nan_to_num(DGMS[:, :, 1] - DGMS[:, :, 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?

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


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