Fractal Wallpaper Patterns

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Douglas Dunham and John Shier: Fractal Wallpaper Patterns. In: Bridges 2015. Pages 183–190



In the past we presented an algorithm that can fill a region with an infinite sequence of randomly placed and progressively smaller shapes, producing a fractal pattern. In this paper we extend that algorithm to fill rectangles and triangles that tile the plane, which yields wallpaper patterns that are locally fractal in nature. This produces artistic patterns which have a pleasing combination of global symmetry and local randomness. We show several sample patterns.

Extended Abstract


 author      = {Douglas Dunham and John Shier},
 title       = {Fractal Wallpaper Patterns},
 pages       = {183--190},
 booktitle   = {Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture},
 year        = {2015},
 editor      = {Kelly Delp, Craig S. Kaplan, Douglas McKenna and Reza Sarhangi},
 isbn        = {978-1-938664-15-1},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{ }},
 url         = { },

Used References

[1] J. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A.K. Peters, Ltd., Wellesley, MA, 2008. ISBN 1-56881-220-5. Wikipedia site for orbifold notation: notation (accessed Apr. 24, 2015)

[2] Doug Dunham and John Shier, The Art of Random Fractals, in Bridges Seoul, (eds. Gary Greenfield, George Hart, and Reza Sarhangi), Seoul, Korea, 2014, pp. 79–86. Also online at:

[3] Christopher Ennis, A Provably “Jam-proof” Algorithm of Filling Space, private communication.

[4] John Shier, Filling Space with Random Fractal Non-Overlapping Simple Shapes ISAMA 2011 Conference Proceedings, page 131, June 13–17, 2011.

[5] Doris Schattschneider, The Plane Symmetry Groups: Their Recognition and Notation, American Mathematical Monthly, 85, 6, 439-450, July, 1978. Wikipedia site for wallpaper groups: group (accessed Apr. 24, 2015)

[6] John Shier and Paul Bourke, An Algorithm for Random Fractal Filling of Space, Computer Graphics Forum, Vol. 32, Issue 8, pp. 89-97, December 2013. Also available on Shier’s web site: (accessed Apr. 24, 2015)

[7] John Shier web site: geom linkpage.html (accessed Apr. 24, 2015)


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