IFS Catalog
A curated collection of Iterated Function System attractors with mathematical definitions and visualizations. 41 entries · 19 tags
Ammann A3 Tiling
Three interleaved tile attractors (A, B, C) forming the Ammann A3 aperiodic quasicrystal tiling. Uses a 4D rational form over ℚ(√5).
Ammann–Beenker Tiling
Two interleaved tile attractors (rhombus and square) forming the Ammann–Beenker 8-fold quasicrystal tiling. Uses a 4D rational form.
Ammann–Beenker Dual
Dual of the Ammann–Beenker tiling via the second octagonal CPS eigenplane. Same 8-fold symmetry, two interleaved fractal tile types.
Antoine's Necklace
24 interlocked tori iterated into a wild Cantor set — homeomorphic to the standard Cantor set but topologically inequivalent to it.
Barnsley Fern
Four affine maps producing a naturalistic fern shape. A classic example of biological structure from simple rules.
Cantor Dust
Four corner maps each scaling by 1/3. The 2D product of two Cantor sets — an uncountable set of isolated points with zero area.
CAP (Hat Monotile)
Canonical Aperiodic Prototile — substitution tiling underlying the hat monotile. Four GIFS attractor types, 30-fold algebraic symmetry.
Cells Tiling
Dual of the Shield tiling via the second CPS projection. Four interleaved fractal tile types with 12-fold symmetry.
Chair Tiling
L-shaped rep-4 tile (L-tromino) generating a nonperiodic limit-periodic tiling with 2-adic cut-and-project structure.
Chinese Lamp
Five copies of the SnowBall fractal surface (13 maps, scale 1/3) assembled into a 3D lamp configuration.
Danzer's 7-Fold Tiling
Three triangular prototiles with 7-fold symmetry. Two variants of the substitution rule. Reference: Nischke & Danzer (1996).
Golden Trapezoid
Four-map algebraic IFS in a 4D rational space; attractor projects onto a golden-ratio trapezoid tiling of the plane.
Gosper Island
Seven self-similar maps with hexagonal algebra. A solid 2D tile related to x²−5x+7.
HD Tile
Self-similar tile with 9 maps, scale 1/3, and boundary dimension ≈ 1.993.
Heighway Dragon
Two maps each rotating by 45° or −135°, producing a dragon-shaped curve that tiles the plane in groups of four.
Jerusalem Cross
Eight-map algebraic IFS in a 4D rational space; attractor has 4-fold symmetry and Hausdorff dimension ≈ 1.786.
Jerusalem Cube
3D extension of the Jerusalem cross — 20 maps at two scales, Hausdorff dimension ≈ 2.53.
Koch Curve
Four affine maps producing a snowflake-like curve with infinite perimeter and finite area.
Koch Snowflake
Seven-map IFS that tiles a solid snowflake region with 6-fold symmetry. Boundary is the Koch curve with dimension log(4)/log(3).
Labyrinth Tiling
Three interleaved fractal tile types with 8-fold symmetry and silver-ratio inflation.
Lévy C Curve
Two maps, each a 45° rotation scaled by 1/√2. The attractor tiles the plane and resembles the letter C at each scale.
Mekhontsev Wedge
Non-trivial 3D convex rep-tile — 8 maps, scale 1/2.
Menger Sponge
3D analog of the Sierpiński carpet — 20-map IFS with Hausdorff dimension ≈ 2.727.
Octagonal Tiling (1225)
Two interleaved fractal tile types with 8-fold symmetry and silver-ratio inflation. A 2-prototile companion to the Labyrinth tiling.
Octahedron Fractal
Six half-scale copies at octahedron vertices — Hausdorff dimension ≈ 2.585.
McWorter's Pentadendrite
Six-map IFS with 5-fold symmetry, expressed in a 4D rational space via the cyclotomic polynomial.
McWorter's Pentigree
Six-map IFS with 5-fold symmetry over the cyclotomic field Q(ζ₁₀), with contraction ratio 1/φ² where φ is the golden ratio.
Pinwheel Tiling
Aperiodic tiling with infinite tile orientations. Three variants: classical right-triangle tile and two fractal single-tile pinwheels.
Quaquaversal Tiling
3D aperiodic tiling with orientations dense in SO(3) — the 3D analog of the pinwheel tiling.
Rauzy Fractal
Tribonacci-based tile: 3 contractions in a 3D rational space projected to the plane.
Robinson Triangles
Two interleaved triangle attractors (acute and obtuse) forming the Penrose rhombus tiling. Uses a 4D rational form.
Shield Tiling
Four interleaved tile types with 12-fold symmetry and dodecagonal inflation. MLD-equivalent to the Socolar tiling.
Sierpiński Carpet
Eight affine maps tiling a square minus its centre. A 2D analogue of the Cantor set and Sierpiński triangle.
Sierpiński Tetrahedron
3D analog of the Sierpiński triangle — 4-map IFS with Hausdorff dimension 2.
Sierpiński Triangle
Three contractions scaling by 1/2, producing a self-similar triangle with zero area.
Sphinx
Rep-4 hexiamond tile. Its nonperiodic tiling is limit-periodic and pure-point diffractive via cut-and-project over a p-adic internal space.
Square Family
Eleven named rep-4 tiles on the square and triangular lattices: Square, Trapeze, Triangle, Chair, L, Sphinx, and others.
Tame Twin Dragon
Two-map self-similar tile based on the Eisenstein-like integer (1+i√7)/2 — the 'tame' twin dragon variant.
Twin Dragon
Two interlocking Heighway Dragon copies form a solid 2D tile. Controlled by the polynomial x²−2x+2.
Vicsek Fractal
Five maps in a cross arrangement — four corners plus centre. Related to critical percolation clusters.
Viper
Single-tile aperiodic IFS with 9 maps and dense orientations. Inflation by 3, irrational symmetry angle.