Figure 4.

Illustration of filtrations built on fullerene C₆₀. Each point or atom in the point cloud data (i.e., coordinates) of the C50 is associated with a common radius r which increases gradually. As the value of r increases, the solid balls centered at given coordinates grow. These balls eventually overlap with their neighbors at certain r values. Simplices indicating such neighborhood information can be defined through abstract r-dependent simplicial complexes, e.g., alpha complexes and Rips complexes. Note that in the last chart, we have removed some atoms to reveal the central void.