# Thomae's function

**Thomae's function**, named after Carl Johannes Thomae, has many names: the **popcorn function**, the **raindrop function**, the **countable cloud function**, the **modified Dirichlet function**, the **ruler function**,^{[1]} the **Riemann function**, or the **Stars over Babylon** (John Horton Conway's name).^{[2]} This real-valued function of a real variable can be defined as:^{[3]}

It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.

Empirical probability distributions related to Thomae's function appear in DNA sequencing.^{[5]} The human genome is diploid, having two strands per chromosome. When sequenced, small pieces ("reads") are generated: for each spot on the genome, an integer number of reads overlap with it. Their ratio is a rational number, and typically distributed similarly to Thomae's function.

For a general construction on arbitrary metric space, see this article