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In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group Γ=PSL(2,ℤ[i]) on . Graphical interpretation of amalgamation of the components of Γ is also given. Some elements of and their conjugates over ℚ(i) have different signs in the orbits of the biquadratic field when acted upon by Γ. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Γα, and they form a closed path which is the only closed path in the orbit Γα. We also devise a procedure to obtain ambiguous numbers of the form , where b is a positive integer.