Some Hyperbolic Fractal Tilings

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Robert W. Fathauer: Some Hyperbolic Fractal Tilings. In: Bridges 2014. Pages 87–94



The concepts of fractal tiling and hyperbolic tiling are combined to create novel fractal surfaces in Euclidean three- space. Paper folding and computer modeling are used to create these constructs. We show examples using triangular, trapezoidal, and dart-shaped prototiles. Smaller tiles deflect out of the plane of adjacent larger tiles, resulting in non- planar surfaces, the shapes of which are dependent on the rules governing the sign of the deflections as well as their magnitudes. These constructs are an intriguing blend of organic and geometric character and in some cases bear marked resemblance to natural leaves.

Extended Abstract


Used References

[1] John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., Wellesley, MA, 2008.

[2] Robert W. Fathauer, Fractal tilings based on kite- and dart-shaped prototiles, Computers & Graphics, Vol. 25, pp. 323-331, 2001.

[3] Robert W. Fathauer, Fractal Tilings Based on Dissections of Polyominoes, Polyhexes, and Polyiamonds, in Homage to a Pied Puzzler, ed. by Ed Pegg, Jr., Alan H. Schoen, and Tom Rodgers, A K Peters, Ltd., Wellesley, MA, 2009.

[4] Edmund Harriss, Maxwell’s Demon website,, accessed February 5, 2014.

[5] Geoffrey Irving and Henry Segerman, Developing fractal curves, J. of Mathematics and the Arts, Vol. 7, pp. 103-121, 2013.

[6] Peichang Ouyang and Robert Fathauer, Aesthetic Patterns Based on Fractal Tilings, IEEE Computer Graphics, Vol. 34, pp 68-76, 2014.

[7] Hans-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe, Fractals for the Classroom, Springer- Verlag, New York, 1992.3

[8] Przemyslaw Prusinkiewicz and Pierre Barbier de Reuille, Constraints of Space in Plant Development, J. of Experimental Botany, Vol. 61, pp. 2117-2129, 2010.

[9] Daina Taimina, Crocheting Adventures with Hyperbolic Planes, A K Peters, Ltd., Wellesley, MA, 2009.


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