The chromatic number of Euclidean space with dense color classes

In this note we construct colorings of Euclidean space $\mathbb{R}^n$ with finitely many colors such that any two points at unit distance have different colors and, in addition, each color class is dense in $\mathbb{R}^n$.

In particular, 12 dense colors suffice to color $\mathbb{R}^2$.

In arbitrary dimension, we show that $n\chi(\mathbb{R}^n)+1$ colors suffice, where $\chi(\mathbb{R}^n)$ denotes the chromatic number of $\mathbb{R}^n$ in the standard formulation.