“Mathematicians have theoretically mapped out the regular division of a plane because this is part of crystallography. Does it therefore belong exclusively to mathematics? I do not think so …”• Escher on Escher: Exploring the Infinite
Tessellation, or tiling, is created when a shape is repeated many times covering a flat area without any overlaps or openings and was clearly Escher's primary interest; so much so that in 1958 he devoted an artist's book entirely to this subject. It was a fascination that began after he stopped producing Italian landscapes around 1937–38. Searching for ways to fill an entire sheet with recognizable and identical forms, he studied the laws of geometry and drew inspiration from Moorish mosaics and articles on crystallography.
On his first visit in 1922, Escher was greatly impressed by the Moorish mosaics decorating the Alhambra in Granada and the Mosque at Cordoba, both vestiges of medieval Islamic Spain. On a second trip, in 1936, he spent several days copying the decorations and became intrigued by the possibility of replacing the purely geometric forms with recognizable elements.
After several intuitive efforts, in October 1937 Escher began to take a more abstract and theoretical approach. Guided by his half-brother Beer, a geology professor, he consulted numerous articles on crystallography, a science related to mineralogy that studies crystalline forms, examining their structure, formation and properties.
As early as 1922, Escher was intrigued by repeating patterns that used interlocking human forms. However it was not until after he had seen Moorish mosaics for the second time in Spain in 1936, and had stopped making prints of Italian landscapes a year later, that he began to seriously develop plane-filling motifs.
In 1958, he wrote and illustrated a book titled The Regular Division of the Plane, published by the De Roos Foundation in Utrecht. In order to fill the plane with recognizable and interlocking geometric shapes, Escher's figures undergo changes of shape, which are detailed in his book.
These three animations created from the artist’s artwork were inspired by his detailed writings. They provide insight on how the plane is divided to create the underlying structure.
In 1957, the Canadian mathematician H. S. M. Coxeter sent Escher one of his articles on hyperbolic geometry. Escher was greatly inspired by one of the illustrations. He had always been frustrated by the boundary imposed by the edge of the paper, which limited the tiling and cut off the recognizable forms.
Coxeter's geometric representation offered a solution and his model enabled Escher to suggest the infinitely small in his work.
Applying geometric principles to his representation of infinity, Escher proceeded by similarity, which may be simply defined as a transformation changing the size but not the shape of the figure. The figures remain identical, but their size can be reduced. Escher made a number of prints based on this principle.