The Family Tree of Fractal Curves
A taxonomy of plane-filling curves using complex integer lattices
by Jeffrey Ventrella
1 Introduction
2 Taxonomy
3 Iteration
4 Morphology
5 Family Resemblances
6 Undiscovered Curiosities
7 References




References



[1] Abelson, H., and diSessa, A. Turtle Geometry. MIT Press, 1986.

[2] Arndt. J. Plane-filling Curves On All Uniform Grids. http://informatique.umons.ac.be/algo/csd8/slides/Arndt_CSD8_long.pdf ( https://arxiv.org/pdf/1607.02433.pdf )

[3] Bader, Michael. Space-Filling Curves - An Introduction with Applications in Scientific Computing. Springer Science and Business Media, 2012.

[4] Bandt, C. Self-similar tilings and patterns described by mappings. Mathematics of Aperiodic Order (ed. R. Moody) Proc. NATO Advanced Study Institute C489, Kluwer 1997, 45-83.

[5] Bandt, C. Mekhontsev, D. and Tetenov, A. A single fractal pinwheel tile. Proc. Amer. Math. Soc. 146, 1271–1285 (2018).

[6] Bandt, C. Personal communication, 2017.

[7] Bogomolny, A. Plane Filling Curves: All Peano Curves. https://www.cut-the-knot.org/Curriculum/Geometry/PeanoComplete.shtml

[8] Chang, A. and Zhang, T. On the Fractal Structure the Boundary of Dragon Curve: http://www.coiraweb.com/poignance/math/Fractals/Dragon/Bound.html

[9] Davis, C. and Knuth, D. Number Representations and Dragon Curves. Journal of Recreational Mathematics. 3 (1970), 66-81, 133-149

[10] Dekking, Michel. Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles. Theoretical Computer Science, 414, (2012) 20-37

[11] Fathauer, R. Fractal art, published on https://www.mathartfun.com/robertfathauer.com/index.html

[12] Fiedorowicz, Z. . √4 triangle grid family triangle curve variation, published at: http://www.math.osu.edu/~fiedorow/math655/examples2.html

[13] Fukuda, M. Shimizu and G. Nakamura, New Gosper Space Filling Curves, Proceedings of the International Conference on Computer Graphics and Imaging (CGIM2001) 34--38 . 2001

[14] Gilbert. W. Fractal Geometry Derived from Complex Bases. The Mathematical Intelligencer. vol. 4. 1982

[15] Goucher. A. (personal communication) Blog post: Complex Projective 4-Space blog: https://cp4space.wordpress.com/

[16] Haverkort, H. Three pretty plane-filling curves, December, 2016. available online at: http://herman.haverkort.net/lib/exe/fetch.php?media=research:pretty-curves.pdf

[17] Hilbert, D. "Über die stetige Abbildung einer Linie auf ein Flächenstück", Math. Ann., 1891 (38), pp. 459-460.

[18] Hutchinson, John, E. Fractals and Self-Similarity. Indiana University Mathematics Journal. Vol. 30, No. 5 (September–October, 1981), pp. 713-747

[19] Irving, G. and Segerman, H. Developing Fractal Curves. Journal of Mathematic in the Arts. Vol 6, 2013, Issue 3-4

[20] Karzes, T. Tiling fractal curves published online at: http://www.karzes.com/xfract/xfract.html

[21] Kayne, Brian. H. A Random Walk Through Fractal Dimensions. John Wiley & Sons, Jul 11, 2008 - Technology & Engineering

[22] Mandelbrot, B. The Fractal Geometry of Nature. W. H. Freeman and Company. 1977

[23] McKenna, Douglas, M. SquaRecurves, E-Tours, Eddies and Frenzies: Basic Families of Peano Curves on the Square Grid, In: Guy, Richard K., Woodrow, Robert E.: The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History, pp. 49-73, Mathematical Association of America, 1994

[24] Moore, E.H., On Certain Crinkly Curves, Trans Amer. Math Soc., 1, 72-90 (1900)

[25] Peano, G. "Sur une courbe, qui remplit toute une aire plane", Mathematische Annalen 36 (1): 157–160. 1890

[26] Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic Beauty of Plants. Springer, 1990

[27] Radin. C (May 1994). “The Pinwheel Tiling of the Plane”. Annals of Mathematics. 139 (3): 661-702. 2007-10-25.

[28] Ryde, K. Draft papers: “Iterations of the Dragon Curve” ( https://download.tuxfamily.org/user42/dragon/dragon.pdf ), 2017; “Iterations of the R5 Dragon Curve” (https://download.tuxfamily.org/user42/r5dragon/r5dragon.pdf) 2018; “Iterations of the Terdragon Curve” (https://download.tuxfamily.org/user42/terdragon/terdragon.pdf), 2018.

[29] Sagan, Hans. Space-filling Curves. Springer Science and Business Media, 2012.

[30] Schraa, W. Range Fractal, published online at: http://wolter.home.xs4all.nl/index.html

[31] Stange, K. Visualizing the Arithmetic of Imaginary Quadratic Fields. International Mathematics Research Notices, Volume 2018, Issue 12, 13 June 2018, Pages 3908–3938

[32] Steeman, Dieter. K. personal communication ( http://demonstrations.wolfram.com/author.html?author=Dieter+Steemann )

[33] Teachout, G. Spacefilling curve designs featured on the web site: http://teachout1.net/village/

[34] Tricot, Claude. Curves and Fractal Dimension. Springer Science & Business Media, Nov.18, 1994. Mathematics.

[35] Ventrella, J. Brainfilling Curves - A Fractal Bestiary. Eyebrain Books/Lulu Press, 2012

[36] Ventrella, J. Portraits from the Family Tree of Plane-filling Curves. Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture. 2019.

[37] Weisstein, E. W. “Peano-Gosper Curve”. MathWorld. October, 31, 2013





Figure 7. A self-avoiding curve of the E(2,1)2 family


The Family Tree of Fractal Curves
A taxonomy of plane-filling curves using complex integer lattices
by Jeffrey Ventrella
1 Introduction
2 Taxonomy
3 Iteration
4 Morphology
5 Family Resemblances
6 Undiscovered Curiosities
7 References