Fractals
Allegria fractal and mathematically inspired jewelry.
Apollonian Gasket, a fractal circle packing formed by packing smaller circles into each triangular gap formed by three larger circles. From MathWorld.
Area of the Mandelbrot set. One can upper bound this area by filling the area around the set by disks, or lower bound it by counting pixels; strangely, Stan Isaacs notes, these two methods do not seem to give the same answer.
Balanced ternary reptiles, Cantor's hourglass reptile, spiral reptile, stretchtiles, trisection of India, the three Bodhi problem, and other Fractal tilings by R. W. Gosper.
Circle fractal based on repeated placement of two equal tangent circles within each circle of the figure. One could also get something like this by inversion, starting with three mutually tangent circles, but then the circles at each level of the recursion wouldn't all stay the same size as each other.
The Curlicue Fractal, Fergus C. Murray.
The Dynamic Systems and Technology Project at Boston Univ. offers several Java applets and animations of fractals and iterated function systems.
Expansions geometric pattern creation techniques by John S. Stokes III.
The fractal art of Wolter Schraa. Includes some nice reptiles and sphere packings.
A fractal beta-skeleton with high dilation. Beta-skeletons are graphs used, among other applications, in predicting which pairs of cities should be connected by roads in a road network. But if you build your road network this way, it may take you a long time to get from point a to point b.
Fractal broccoli. Photo by alfredo matacotta. See also this French page.
Fractal geometry and complex bases. Publications and software by W. Gilbert.
Fractal instances of the traveling salesman problem, P. Moscato, Buenos Aires.
Fractal knots, Robert Fathauer.
Fractal patterns formed by repeated inversion of circles: Indra's Pearls Inversion graphics gallery, Xah Lee. Inversive circles, W. Gilbert, Waterloo.
Fractal patterns in the real world, Ian Stewart.
Fractal planet and fractal landscapes. Felix Golubov makes random triangulated polyhedra in Java by perturbing the vertices of a recursive subdivision.
Fractal reptiles and other tilings by IFS attractors, Stewart Hinsley.
Fractal resources. A collection of web links by John Mathews.
Fractals. The spanky fractal database at Canada's national meson research facility.
Generating Fractals from Voronoi Diagrams, Ken Shirriff, Berkeley and Sun.
Geometric Arts. Knots, fractals, tesselations, and op art. Formerly Quincy Kim's World of Geometry.
IFS and L-systems. Vittoria Rezzonico grows fractal broccoli and Sierpinski pyramids.
Sándor Kabai's mathematical graphics, primarily polyhedra and 3d fractals.
Robert F. Kauffman's fractal and Escherian art, with Escher-like animated animal-form tilings.
Labyrinth tiling. This aperiodic substitution tiling by equilateral and isosceles triangles forms fractal space-filling labyrinths.
Mathematical imagery by Jos Leys. Knots, Escher tilings, spirals, fractals, circle inversions, hyperbolic tilings, Penrose tilings, and more.
Line fractal. Java animation allows user control of a fractal formed by repeated replacement of line segments by similar polygonal chains.
Number patterns, curves, and topology, J. Britton. Includes sections on the golden ratio, conics, Moiré patterns, Reuleaux triangles, spirograph curves, fractals, and flexagons.
Paperfolding and the dragon curve. David Wright discusses the connections between the dragon fractal, symbolic dynamics, folded pieces of paper, and trigonometric sums.
Pleats, twists, and sliceforms. Some links to Richard Sweeney's fractal paperfolding art, via dataisnature.
Programming for 3d modeling, T. Longtin. Tensegrity structures, twisted torus space frames, Moebius band gear assemblies, jigsaw puzzle polyhedra, Hilbert fractal helices, herds of turtles, and more.
Rational maps with symmetries. Buff and Henriksen investigate rational functions invariant under certain families of Möbius transformations, and use them to generate symmetric Julia sets.
Reproduction of sexehexes. Livio Zucca finds an interesting fractal polyhex based on a simple matching rule.
Vittoria Rezzonico's Java applets. Hypercube and polyhedron visualization, and circle inversion patterns. Requires both Java and JavaScript.
Sierpinski carpet on the sphere. From Curtis McMullen's math gallery.
The Sierpinski Tetrahedron, everyone's favorite three dimensional fractal. Or is it a fractal?
sneJ made a Mandelbrot set with sheet plastic and a laser cutter.
Spherical Julia set with dodecahedral symmetry discovered by McMullen and Doyle in their work on quintic equations and rendered by Don Mitchell. Update 12/14/00: I've lost the big version of this image and can't find DonM anywhere on the net -- can anyone help? In the meantime, here's a link to McMullen's rendering.
3D strange attractors and similar objects, Tim Stilson, Stanford.
What happens when you connect uniformly spaced but not dyadic rational points along the Peano spacefilling curve? R. W. Gosper illustrates the results.
Yantram sacred art toolbox. Software for creating various kinds of symmetric fractal mandala.
