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| ART AND MATH OF THE INFINITE LINE |
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Hello infinite curly line lovers. I have a long personal history with fractal curves,
starting in 1985 when I wrote my first line of code (to draw the Koch curve).
This web page is a curation of my visual math and geometrical discoveries with fractal curves,
emphasizing those ever-so-clever space-filling curves.
I've included an interactive app for exploring fractal curves, a gallery
featuring my fractal curve art, and some instructional videos and animations
I've made over the years. There are also links to some of my explorations in
identifying an algebraic structure for classifying planefilling curves. At the end
are links to related web sites. Enjoy!
-Jeffrey |
FEATURED VIDEO
A video about how "flipping babies" creates self-avoiding curves |
| WHAT ARE FRACTAL CURVES? |
Meditate on the mandala-like sweep of this curve. |
A curve that bends and curls at every level of magnification is a fractal curve.
It has a fractional dimension between 1 and 2, A curve which is so curvy that
it essentially visits every point in a planar area is a spacefilling curve,
and it defines a continuous mapping from a lower-dimensional space (a line)
into a higher-dimensional space (a region of the plane). Its dimension is 2.
There are infinitely many ways that fractal curves can be crafted so that they fill space. These ways can be expressed in elegant geometrical rules, as demonstrated by Koch Construction, the application of L-systems to Turtle Geometry, and Iterated Function Systems. Viewing a space-filling curve with only a few levels of fractal recursion reveals the beautiful logic of its convoluted path. This page is dedicated to the awesome world of fractal curves, and the clever ways a curly curve can fill space! |
EXPLORE SPACEFILLING CURVES
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| CHAPTERS, BOOKS, ETC. |
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The Menagerie of Plane-filling Fractal Curves: From Visual Analysis to Number Theory
J. Ventrella. 2026. To be published as a chapter in the upcoming book: Handbook of the Mathematics of the Arts and Sciences, ed. Bharath Sriraman, Springer Nature, 2026 Designing Fractal Curves with Five-Fold Rotational Symmetry Using the Complex Number Golden Ratio J. Ventrella. 2021. Proceedings of Bridges: Mathematics, Art, Music, Architecture, Culture The Family Tree of Fractal Curves - A Taxonomy of Planefilling Curves Using Complex Integer Lattices J. Ventrella. 2019, Eyebrain Books/Lulu.com Portraits from the Family Tree of Plane-Filling Curves J. Ventrella. 2019. Proceedings of Bridges: Mathematics, Art, Music, Architecture, Culture Brainfilling Curves - A Fractal Bestiary J. Ventrella. 2012, Eyebrain Books/Lulu.com |
| GALLERIES |
Artful Designs by JJ Ventrella
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A collection of high-res fractal curve designs on the Internet Archive
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| CLASSIFICATION |
Based on complex integers (Gaussian and Eisenstein)
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| OTHER STUFF |
The Dragon of Eve |
Brainfilling Curves |
Branching Patterns |
Branching Koch Snowfale |
Growing Fractal Curves |
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RELATED |
Hyperbolic Kaleidoscopes |
Developing F Curves |
Blue-Brown Hilbert |
Uncommon Fractals |
Fractal curve blending |
The Classic Dragon Curve |
The Levy C Curve |
The Koch Snowflake |
Henry Segerman Video |
Connected Fermat Spirals |
The Gosper Curve |
The Sierpinski Curve |
Doug McKenna's Art |
Fractal Pinwheel Tile |
Jorg Arndt curves |
Smooth curves by Craig Kaplan |
Apollonian Gasket |
Dragon Curve - paper strips
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Nelson Max's film from 1972 |
Spacefilling Curves |
Space-filling Trees |
David Mitchell's Lattice Labyrinths |
spacefillingcurves.org |
McWorter's Pentigree |
Generalized Gosper Curves |
Sierpinski Arrowhead |
Woolly Thoughts |
Fractal Tilings |
Designs by Tony Hammer |
Creating Classic Fractals |
Victor Carbajo Fractal Art |
Wolter Schraa Fractal |
F Curves, Dimension |
Peano Curves |
Teachout Space-filling Curves |
Sagan's book |
Aubrey Jaffer |
Nasu Dengaku |
Mark Dow's Curves |
Henry Segerman |
Fractal Materials |
Dragon Curves, Jos Leys |
Gary's Dragons |
Radix Fractals - Gilbert |
Context-based Curves |
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© 2026 Jeffrey Ventrella 
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