Tessellations from Group Actions and the Mystery of Escher's Solid

Ioana Mihaila
Renaissance Banff: Mathematics, Music, Art, Culture (2005)
Pages 329–330


In the paper [1], Joyce Frost and Peg Cagle show how to construct a tessellation with squares of the plane from another one of smaller squares, and how this process can be generalized in three dimensions to construct a tessellation of the space with rhombic dodecahedra from a tessellation with cubes. The authors then proceed and explain how to construct a stellated rhombic dodecahedron (Escher’s solid), and why this solid is space-filling as well. Interestingly, the procedure of constructing a tessellation from a given one by conveniently cutting some of the tiles can be iterated in all these cases, and the tessellations obtained can be associated to group action in the plane, or three-dimensional space, respectively.