The Family Tree of Fractal Curves
(A taxonomy of plane-filling curves using complex integer lattices)
by Jeffrey Ventrella

(view for free on the Archive)

(view the web version for free)

(buy the book on
Designing Fractal Curves with Five-Fold Rotational Symmetry
by Jeffrey Ventrella

(a paper published in Bridges, 2021)
fractal curves

Realtime fractal curve explorer

Taxonomy of plane-filling curves

collection of high-res designs on the Internet Archive

Gallery of fractal curve designs

     Specimen of the moment    (100 images: changes every 10 seconds)    Select to enlarge
A curve that bends and curls at every level of maginifation is a fractal curve. It has a fractional dimension between 1 and 2, A curve which is so curvey 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 plane). Its dimension is 2. The fascinating thing about these curves is that they are self-similar and tiling by nature. 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 infinite ways a single curly line can fill the space between 1 and 2 dimensions.

Web sites about fractal curves and related subjects

The Classic Dragon Curve The Levy C Curve The Koch Snowflake from MathWorld

The Sound of Space-filling Curves The Dragon of Eve Connected Fermat Spirals

The Gosper Curve The Sierpinski Curve Doug McKenna's Art

Fractal Art bt Robert Fathauer Fractal Art by Tony Hanmer Antonio Marquez-Raygoza's huge web page on the Sierpinksi Triangle

Fractal Pinwheel Tile by Bandt, Mekhontsev, and Tetenov Jorg Arndt: Plane-filling curves on all uniform grids Smooth curves by Craig Kaplan

One of Pat Forster's fractal quilts, inspired by Ventrella's Dragon of Eve Apollonian Gasket by alunw Dragon Curve made with folded paper strips

Grammatical Evolution Cyclomers by Stewart Hinsley Nelson Max's film from 1972

Spacefilling Curves Space-filling Trees Branching Patterns McWorter's Pentigree Fractal circle art from fdecomite

Generalized Gosper Curves (Jin Akiyama, Hiroshi Fukuda, Hiro Ito, and Gisaku Nakamura) Boundary of the Dragon Curve (Chang/Zhang) Scott Kim's Fractal Tree

The Sierpinski Arrowhead Curve Dragon Curves from Woolly Thoughts Bowyer Curve

Cool Curves from Neuro Fuzzy Compendium of Fractal Tilings Fractal Designs by Tony Hammer

Aperiodic Tilings by Mark McClure Creating Classic Fractals "Impossible" Fractals

Victor Carbajo's L-Systems Fractal Art Applications for Space-Filling Curves Wolter Schraa's Range Fractal

Fractal Curves and Dimension, from "Cut the Knot" Peano Curves (from The Geometry Center) Gary Teachout's Space-filling Curves

Hans Sagan's book on Space-Filling Curves Aubrey Jaffer's Multi-Dimensional Space-Filling Curves Nasu Dengaku's Space-Filling Curve Art

Tom Karzes' Tiling Fratal Curves Fractal jewelry by Dragon Nerd Mark Dow's Beautiful Curves

Space-filling curves for spatial data structures (Herman Haverkort & Freek van Waldervee) Space-filling Curves (from Mark C. Chu-Carroll, in Good Math, Bad Math) Beautiful Images by Miqel

The amazing mathematical art of Henry Segerman Fractal Materials Dragon Curves by Jos Leys

Random Spacefilling by Paul Bourke Tesselation and Spacefilling Gary's Dragons

Radix Fractals by William Gilbert Schmidt Arrangements by Katherine E. Stange Branching Snowflake

Context-based Space-filling Curves bending Hyperbolic Kaleidoscopes Smooth Koch

Cool images by Richard Lewis Henry Segerman Video on Fractal Curves David Mitchell's Lattice Labyrinths

Videos about Fractal Curves

Brainfilling Curves

Fractal Folding

The Dragon of Eve

Vi Hart Squiggles

Segerman's talk

Fractal charm: Space filling curves

Dragon Fractal - Math Guy

November Colormap

Growing the Arrowhead Curve

All Dragon Curves

Growing the Snowflake Sweep

Unfolding the Dragon

This web site was created by Jeffrey Ventrella. Email "jeffrey at ventrella dot com" to suggest other links to add.