A kaleidoscope is obtained as the quotient of a space by the discontinuous action of a discrete group of transformations; this can also be obtained from a fundamental domain which characterizes it. In the present study, the specific case of the Hyperbolic Plane is analyzed with respect to the action of a hyperbolic polygonal group, which is a particular case of an NEC group. Under the action of these groups, the hyperbolic plane is tessellated using tiles with a polygonal shape. The generators of the group are reflections in the sides of the polygon. Clear examples of quadrilateral tessellations of the hyperbolic plane with Saccheri and Lambert quadrilaterals -designed using the Hyperbol package created for Mathematica software- and are found in the basic structure of some of the mosaics of M.C. Escher.