Hidden Beauty in Penrose Tiling: Weavings & Lace

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Douglas Burkholder: Hidden Beauty in Penrose Tiling: Weavings & Lace. In: Bridges 2017, Pages 213–220.



We explore Penrose’s tiling using kites and darts in search for hidden beauty. We focus on the iterative subdivision process that can be used to create Penrose tilings by subdividing half kites and half darts into smaller half kites and half darts. By selectively coloring the half darts and half kites, based only on their relative position in the subdivision process, we create 15 unexpected and distinctive patterns hidden within Penrose tiling. These patterns tend to have the appearance of a weaving. Alternately, by selectively discarding tiles as we recursively subdivide, we can obtain fractal patterns that appear lacelike. We finish by filling the negative spaces in the fractal patterns with pursuit curves.

Extended Abstract


 author      = {Douglas Burkholder},
 title       = {Hidden Beauty in Penrose Tiling: Weavings \& Lace},
 pages       = {213--220},
 booktitle   = {Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture},
 year        = {2017},
 editor      = {David Swart, Carlo H. S\'equin, and Krist\'of Fenyvesi},
 isbn        = {978-1-938664-22-9},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{http://archive.bridgesmathart.org/2017/bridges2017-213.pdf}}

Used References

[1] Beautiful Penrose Tile in Architecture: http://www.eschertile.com/penrose.htm

[2] D.G. Burkholder. Unexpected Beauty Hidden in Radin-Conway’s Pinwheel Tiling. In Bridges Baltimore 2015: Mathematics, Music, Art, Architecture, Culture: 383-386. 2015.

[3] B. Grünbaum and G.C. Shephard, Tilings and Patterns, W.H. Freeman. 1987.

[4] D. Reimann. Binomial Pursuit. MAA Mathematics Magazine: 89 (3). 2016.

[5] The Tiling Encyclopedia: http://tilings.math.uni-bielefeld.de/


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