  A finite subset of a Euclidean space is called an $s$-distance set if there
exist exactly $s$ values of the Euclidean distances between two distinct points
in the set. In this paper, we prove that the maximum cardinality among all
5-distance sets in $\mathbb{R}^3$ is 20, and every $5$-distance set in
$\mathbb{R}^3$ with $20$ points is similar to the vertex set of a regular
dodecahedron.
