Let the Numbers Do the Walking: Generating Turtle Dances on the Plane from Integer Sequences

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Adam Colestock: Let the Numbers Do the Walking: Generating Turtle Dances on the Plane from Integer Sequences. In: Bridges 2017, Pages 139–146.



I tell a story of a dancing turtle and introduce a mathematical form based in turtle geometry and modular arithmetic. Sequences of integers generated from a discrete parametric function determine the turn angles for a single meandering point and produce symmetric and varied designs on the plane. I also analyze the mathematical properties of the form, highlight emergent features within the form set and show designed objects incorporating these patterns. Finally, I propose several variations of the initial algorithm that hold promise for future inquiry and mathematics-driven making.

Extended Abstract


 author      = {Adam Colestock},
 title       = {Let the Numbers Do the Walking: Generating Turtle Dances on the Plane from Integer Sequences},
 pages       = {139--146},
 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/bridges2017-139.pdf}}

Used References

[1] Papert, Seymour. Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc., 1980.

[2] Abelson, Harold, and Andrea A. DiSessa. Turtle geometry: The computer as a medium for exploring mathematics. MIT press, 1986.

[3] Turtle Art. https://turtleart.org/ (as of January 30, 2017)

[4] Gailiunas, P. (2005). Meanders. In Renaissance Banff: Mathematics, Music, Art, Culture, pages 25-30. Bridges Conference. Available online at http://archive.bridgesmathart.org/2005/bridges2005-25.pdf

[5] Zantema, Hans. Turtle graphics of morphic sequences. Fractals 24.01 (2016): 1650009.

[6] Verhoeff, Tom. 3D turtle geometry: artwork, theory, program equivalence and symmetry. International Journal of Arts and Technology 3.2-3 (2010): 288-319.

[7] Reas, Casey, and Ben Fry. Processing: a programming handbook for visual designers and artists. No. 6812. MIT Press, 2007.

[8] Chappell, David. Sinuous Meander Patterns in Natural Coordinates. In Robert Bosch, Douglas McKenna and Reza Sarhangi, editors, Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, pages 183-190, Phoenix, Arizona, 2012, Tessellations Publishing. Available online at http://archive.bridgesmathart.org/2012/bridges2012-183.html

[9] Ramachandran, Vilayanur S., and Edward M. Hubbard. "Synaesthesia--a window into perception, thought and language." Journal of consciousness studies 8.12 (2001): 3-34.

[10] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. http://oeis.org (as of January 30, 2017)


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