Abstract
In this paper we study some combinatorial properties of biquadratic irrational number field , under the action of a Bianchi group . In this experiment it is revealed that a special class of elements exists; that is, for an element its conjugate has different signs in the closed path (orbits) for the action of over , known as ambiguous numbers. It is also proved that the orbit defined on a finite number of ambiguous numbers succeeding a unique closed path.