The mathematical investigations of Roger Penrose

in his drawings the constructive elements of this world. We introduce unusual higher-dimensional computer reconstructions of some drawings for presenting our research results in solving geometric modeling problems arising from analysis of Escher’s artworks. The following identified problems had no general solutions in geometric modeling: the metamorphosis between 2D arbitrary polygons similar to shape transformations in “Day & Night ” (1938), the simulation of 3D shape reconstruction from its 2D projection as it happens in “Reptiles ” (1943), and the surface trimming depicted in the “Rind ” (1955) and other Escher’s drawings. We applied the Function Representation (FRep) of geometric shapes by continuous functions of point coordinates [1, 4] to solve these problems. A visually smooth transformation (or metamorphosis) between two arbitrary shapes can find a general solution if both shapes are represented by real functions and the transformation is described as an interpolation between these functions. First, we represent a 2D polygon by a real function of point coordinates taking zero value at polygon edges. An arbitrary polygon can be represented by a settheoretic expression involving union and intersection operations on the half-planes passing through the polygon edges. The formula for the function defining a polygon can be obtained from the settheoretic expression by replacing each half-plane by its defining function while set-theoretic operations are replaced by the corresponding R-functions, see details in [2]. We implemented the above polygon-to-function conversion approach and applied it to several geometric modeling problems including shape metamorphosis and reconstruction from projections. The original Escher’s drawing and our reconstruc