isPure tests if the Fan/PolyhedralComplex is pure by checking if the first and the last entry in the list of generating Cones/Polyhedra are of the same dimension.
Let us construct a fan consisting of the positive orthant and the ray v that is the negative sum of the canonical basis, which is obviously not pure:
i1 : C = posHull matrix {{1,0,0},{0,1,0},{0,0,1}}
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o1 : Cone
i2 : v = posHull matrix {{-1},{-1},{-1}}
o2 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 1
number of facets => 1
number of rays => 1
o2 : Cone
i3 : F = fan {C,v}
o3 = {ambient dimension => 3 }
number of generating cones => 2
number of rays => 4
top dimension of the cones => 3
o3 : Fan
i4 : isPure F
o4 = false
But we can make a pure fan if we choose any two dimensional face of the positive orthant and take the cone generated by this face and v and add it to the cone:
i5 : C1 = posHull{(faces(1,C))#0,v}
o5 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o5 : Cone
i6 : F = addCone(C1,F)
o6 = {ambient dimension => 3 }
number of generating cones => 2
number of rays => 4
top dimension of the cones => 3
o6 : Fan