A Meditation on Kepler's Aa

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Craig S. Kaplan: A Meditation on Kepler's Aa. In: Bridges 2006. Pages 465–472



Kepler’s Harmonice Mundi includes a mysterious arrangement of polygons labeled Aa, in which many of the poly- gons have fivefold symmetry. In the twentieth century, solutions were proposed for how Aa might be continued in a natural way to tile the whole plane. I present a collection of variations on Aa, and show how it forms one step in a sequence of derivations starting from a simpler tiling. I present alternate arrangements of the tilings based on spirals and substitution systems. Finally, I show some Islamic star patterns that can be derived from Kepler-like tilings.

Extended Abstract


Used References

[1] Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy. Unsolved Problems in Geometry. Springer- Verlag, 1991.

[2] L. Danzer, B. Gr ̈unbaum, and G. C. Shephard. Can all tiles of a tiling have five-fold symmetry? Amer- ican Mathematical Monthly, 89:568–585, 1982.

[3] Branko Gr ̈unbaum and G. C. Shephard. Tilings and Patterns. W. H. Freeman, 1987.

[4] Craig S. Kaplan. Computer Graphics and Geometric Ornamental Design. PhD thesis, Department of Computer Science & Engineering, University of Washington, 2002.

[5] Craig S. Kaplan. Islamic star patterns from polygons in contact. In Proceedings of the 2005 conference on graphics interface. Canadian computer human communication society, 2005.

[6] Roger Penrose. Pentaplexity. Mathematical Intelligencer, 2:32–37, 1979/80.

[7] John Savard. Pentagonal tilings. http://www.quadibloc.com/math/penint.htm


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