# 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.