# Čech Complex Playground

Interactive tools for understanding nerves

Čech complexes are one of the main tools for building a simplicial complex from data and are used regularly for Persistent Homology. This post provides an interactive demonstration of the Čech complex for a 2-dimensional data set. It is meant to help newcomers gain an intuition about the complex.

### Visualization

The resulting simplicial complex is overlaid on top of the covers. Here we only calculate the nerve up to order 2, i.e. vertices, edges, and triangles.

- the slider lets you change the radius of the balls and
- the data points can be rearranged by dragging them.

### Mathematical Background

The Čech complex of a data set \(X\) with radius \(r\) is defined as the simplicial complex

$$ Č(X) = \big\{ \sigma_I \mid \bigcup_{i \in I} B_r(x_i) \neq \emptyset \big\} $$

All this really says is we add a simplex \(\sigma_I\) on vertices \( \{ x_i\}_{i\in I} \) when balls with radius \(r\) around each vertex all have a nonempty intersection.

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