Recognizable Motif Tilings Based on Post-Escher Mathematics
Year: 1999 Authors: Robert W. Fathauer
Core claim
Post-1972 mathematical ideas, especially Penrose tiles and fractals, enable recognizable-motif tilings with multiple solution types and measurable self-similarity.
Topics
recognizable motif tilings, Penrose tiles, fractal design, self-similarity
Domains
tessellation, nonperiodic tilings, fractal dimension, box counting, print design, pattern design, visual art, motif-based tiling
Methods
matching rules, template-based construction, iterative process, box counting method
Media
rhombi, tile motifs, prints, sea life motif, dragon motif
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
BRIDGES Mathematical Connections in Art, Music, and Science
Recognizable Motif Tilings Based on Post-Escher Mathematics
Robert W. Fathauer Tessellations 688 W. 1st Street., Ste. 5 Tempe, AZ 85281 Tessella@futureone.com
M.C. Escher was preoccupied for most of his career as an artist with the covering of the Euclidean plane by tiles with recognizable motifs, generally lizards, birds, and the like. In his notebooks, 137 such designs are enumerated, and several of these were used as the bases for some of his best-known finished prints. All of these sets of tiles only fit together one way.
New tilings with recognizable motifs have been designed based on mathematical discoveries made around the time of Escher’s death in 1972. In particular, the nonperiodic Penrose and related tiles, and the concept of fractals are employed.
In the case of Penrose and related tiles, the edges of rhombi are replaced with a single line segment in different aspects. First, a pair of rhombi with interior angles and is considered. Of the numerous distinct ways in which a line segment can be placed in the eight locations (defining matching rules), some do not tile at all, at least one only allows nonperiodic tilings (the Penrose set P2), and some allow both nonperiodic and periodic tilings. The set P2 allows an infinite number of distinct tilings, and recognizable-motif tiles based on this set have been demonstrated. A set of matching rules which allows both nonperiodic and periodic tilings is used to form tiles with a sea life motif. Generalized Penrose rhombi for and (cf. 5) and the Penrose set P1 are also explored and used as templates for multiple-solution recognizable-motif tilings.
The concept of fractals was not well developed or widely known during Escher’s lifetime. Even so, several of his prints possess some fractal character. These prints, in which infinite tilings of the Euclidean plane are represented in finite areas have been divided into three categories by Bruno Ernst: square-division prints, spiral prints, and the Coxeter prints, which employ hyperbolic geometry. These are fractal in the sense that they exhibit self-similarity on different scales; however, none of the tiles or perimeters of these prints are fractal. Original recognizable-motif tiling prints have been devised which have different types of fractal character. A design with a fractal perimeter and a non-fractal seal motif is one example. Two designs with fractal tiles have been executed in which singularities are distributed throughout the designs, not just at the perimeters. Finally, a self-replicating fractal tile is demonstrated which has a dragon motif. This tile is the most fractal of the lot in the sense that it is generated by an iterative process. The box counting method to used to compute a fractal dimension for this tile is approximately 1.5.
Robert W. Fathauer
Figure 1: Fractal Serpents
Figure 2: Fractal Maskes
Figure 3: Bats and Owls