This paper is an attempt for convergence of mathematical and logical thinking and emotional imagination. It will provide teachers with the resources to be applied to a long-term project such as mentoring program or independent study for the gifted students in secondary school.

In this paper, we take four steps to construct hyperbolic tilings. First, we begin with a quick review of circle inversion upon which hyperbolic geometry is based. Second, we make the straightedge and compass in Poincaré disk using script function in Geometric Sketch Pad (GSP) with together the construction tool for inversion points. Third, we analyze Escher’s Circle Limit I-IV using the construction tool for inversion points and the straightedge in Poincaré disk model. Fourth, on the basis of the results analyzing Circle Limits we construct the regular polygonal tilings in Poincaré disk and reconstruct Escher’s Circle Limit II using GSP’s script tools.

tessellation, hyperbolic geometry, Poincaré disk, Escher’s Circle Limits, inversion.