Natural Color Symmetry

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Frank A. Farris: Natural Color Symmetry. In: Bridges 2017, Pages 131–138.



Existing techniques to create images with color-reversing (or other kinds of color symmetry) produce results that look less than natural. We produce images with approximate color symmetries by using the domain coloring algorithm with suitable complex Fourier series and photographs that have varying degrees of naturally occurring color symmetry. The technique yields patterns where the color symmetry looks less crystallographic than those previously produced, but perhaps more artistic. The technique produces an interesting combination of symmetrical and natural patterns.

Extended Abstract


 author      = {Frank A. Farris},
 title       = {Natural Color Symmetry},
 pages       = {131--138},
 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{}}

Used References

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

[2] Frank Farris. Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. Princeton University Press, 2015.

[3] Frank Farris, Review of Visual Complex Analysis, by Tristan Needham. Amer. Math. Monthly, 105: 570–576, 1998.

[4] Frank Farris. Symmetric yet organic: Fourier series as an artists tool. J. Math. Arts, 2013. doi: 10.1080/17513472.2013.819314

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

[6] Son Ngo and Bridget Went. SymmetryWorks software. Available for download at as of January 29, 2017.

[7] Venera Adanova and Sibel Tari. Extracting Unit Cells from Tilings with Color Symmetries: Case of Counterchange Patterns. In Carlo Sequin Douglas McKenna Kristof Fenyvesi Eve Torrence, Bruce Torrence and Reza Sarhangi, editors, Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture, pages 453–456, Phoenix, Arizona, 2016. Tessellations Publishing. Available online at

[8] Dorothy Washburn and Donald Crowe. Symmetries of Culture, Theory and Practice of Plane Pattern Analysis. University of Washington Press, 1988.

[9] H.J. Woods. The Geometrical Basis of Pattern Design Part IV: Counterchange Symmetry in Plane Patterns. Journal of the Textile Institute, Transactions, 27: 305–320, 1936.


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