# Gaussian rational

The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres. In this construction, the complex numbers are embedded as a plane in a three-dimensional Euclidean space, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point. For a Gaussian rational represented in lowest terms as ${\displaystyle p/q}$, the radius of this sphere should be ${\displaystyle 1/q{\bar {q}}}$ where ${\displaystyle {\bar {q}}}$ represents the complex conjugate of ${\displaystyle q}$. The resulting spheres are tangent for pairs of Gaussian rationals ${\displaystyle P/Q}$ and ${\displaystyle p/q}$ with ${\displaystyle |Pq-pQ|=1}$, and otherwise they do not intersect each other.[2][3]