According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
Deterministic fractals Hausdorff dimension(exact value) Hausdorff dimension(approx.) Name Illustration Remarks Calculated 0.538 Feigenbaum attractor The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic map for the critical parameter value , where the period doubling is infinite. This dimension is the same for any differentiable and unimodal function. 0.6309 Cantor set Built by removing the central third at each iteration. Nowhere dense and not a countable set. 0<D<1 1D generalized symmetric Cantor set Built by removing the central interval of length from each remaining interval of length at the nth iteration. produces the usual middle-third Cantor set. Varying between 0 and 1 yields any fractal dimension . 0.6942 (1/4, 1/2) asymmetric Cantor set Built by removing the second quarter at each iteration. (golden ratio). 0.69897 Real numbers whose base 10 digits are even Similar to the Cantor set. 0.88137 Spectrum of Fibonacci Hamiltonian The study of the spectrum of the Fibonacci Hamiltonian proves upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that the spectrum converges to an explicit constant. 1 Smith–Volterra–Cantor set Built by removing the central interval of length from each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of 1/2. 1 Cantor triangle Built by replacing a triangle by three copies, iteratively, with ratio 1/3. For r between 0 and 1/2, the fractal dimension is . 1 Takagi or Blancmange curve Defined on the unit interval by , where is the triangle wave function. Not a fractal under Mandelbrot's definition, because its topological dimension is also . Special case of the Takahi-Landsberg curve: with . The Hausdorff dimension equals for in . (Hunt cited by Mandelbrot). 1.0295240599 Boundary of the supergolden fractal triangle Here is the supergolden ratio (real root of ) and is the real root of . Calculated 1.0812 Julia set z2 + 1/4 Julia set of f(z) = z2 + 1/4. Solution of 1.0933 Boundary of the Rauzy fractal Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: , and .Messaoudi, Ali. Frontième de numération complexe", matwbn.icm.edu.pl. Accessed: 27 October 2018. is one of the conjugated roots of . 1.12915 contour of the Gosper island Term used by Mandelbrot (1977). The Gosper island is the limit of the Gosper curve. Measured (box counting) 1.2 Dendrite Julia set Julia set of f(z) = z2 + i. 1.2083 Fibonacci word fractal 60° Built from the Fibonacci word. See also the standard Fibonacci word fractal. (golden ratio). 1.2108 Boundary of the tame twindragon One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size).Ngai, Sirvent, Veerman, and Wang (October 2000). "On 2-Reptiles in the Plane 1999", Geometriae Dedicata, Volume 82. Accessed: 29 October 2018.Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems", Wolfram.com. 1.26 Hénon map The canonical Hénon map (with parameters a = 1.4 and b = 0.3) has Hausdorff dimension 1.261 ± 0.003. Different parameters yield different dimension values. 1.2619 Triflake Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes. 1.2619 Koch curve 3 Koch curves form the Koch snowflake or the anti-snowflake. 1.2619 boundary of Terdragon curve L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle. 1.2619 2D Cantor dust Cantor set in 2 dimensions. 1.2619 2D L-system branch L-Systems branching pattern having 4 new pieces scaled by 1/3. Generating the pattern using statistical instead of exact self-similarity yields the same fractal dimension. Calculated 1.2683 Julia set z2 − 1 Julia set of f(z) = z2 − 1. 1.3057 Apollonian gasket Starting with 3 tangent circles, repeatedly packing new circles into the complementary interstices. Also the limit set generated by reflections in 4 mutually tangent circles. See McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu. Accessed: 27 October 2018. 1.328 5 circles inversion fractal The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See Chang, Angel and Zhang, Tianrong. pdf 1.36521Mandelbrot, B. B. (1983). The Fractal Geometry of Nature, p.48. New York: W. H. Freeman. . Cited in: Quadratic von Koch island using the type 1 curve as generator Also known as the Minkowski Sausage Calculated 1.3934 Douady rabbit Julia set of f(z) = z2 -0.123 + 0.745i 1.4649 Vicsek fractal Built by exchanging iteratively each square by a cross of 5 squares. 1.4649 Quadratic von Koch curve (type 1) One can recognize the pattern of the Vicsek fractal (above). 1.4961 Quadric cross Built by replacing each end segment with a cross segment scaled by a factor of 51/2, consisting of 3 1/3 new segments, as illustrated in the inset.
Images generated with Fractal Generator for ImageJ. 1.5000 a Weierstrass function: The Hausdorff dimension of the graph of the Weierstrass function defined by with and is .N. Zhang. The Hausdorff dimension of the graphs of fractal functions. (In Chinese). Master Thesis. Zhejiang University, 2018. 1.5000 Quadratic von Koch curve (type 2) Also called "Minkowski sausage". 1.5236 Boundary of the Dragon curve cf. Chang & Zhang.Fractal dimension of the boundary of the dragon fractal 1.5236 Boundary of the twindragon curve Can be built with two dragon curves. One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size). 1.5850 3-branches tree Each branch carries 3 branches (here 90° and 60°). The fractal dimension of the entire tree is the fractal dimension of the terminal branches. NB: the 2-branches tree has a fractal dimension of only 1. 1.5850 Sierpinski triangle Also the limiting shape of Pascal's triangle modulo 2. 1.5850 Sierpiński arrowhead curve Same limit as the triangle (above) but built with a one-dimensional curve. 1.5850 Boundary of the T-square fractal The dimension of the fractal itself (not the boundary) is 1.61803 a golden dragon Built from two similarities of ratios and , with . Its dimension equals because . (golden ratio). 1.6309 Pascal triangle modulo 3 For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram). 1.6309 Sierpinski Hexagon Built in the manner of the Sierpinski carpet, on an hexagonal grid, with 6 similitudes of ratio 1/3. The Koch snowflake is present at all scales. 1.6379 Fibonacci word fractal Fractal based on the Fibonacci word (or Rabbit sequence) Sloane A005614. Illustration : Fractal curve after 23 steps (F23 = 28657 segments).The Fibonacci word fractal (golden ratio). Solution of 1.6402 Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3 Generalization : Providing the open set condition holds, the attractor of an iterated function system consisting of similarities of ratios , has Hausdorff dimension , solution of the equation coinciding with the iteration function of the Euclidean contraction factor: . 1.6667 32-segment quadric fractal (1/8 scaling rule) see also: :File:32 Segment One Eighth Scale Quadric Fractal.jpgBuilt by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 32/log 8 = 1.6667. Images generated with Fractal Generator for ImageJ. 1.6826 Pascal triangle modulo 5 For a triangle modulo k, if k is prime, the fractal dimension is (cf. Stephen Wolfram). Measured (box-counting) 1.7 Ikeda map attractor For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map . It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values. 1.6990 50 segment quadric fractal (1/10 scaling rule) Built by scaling the 50 segment generator (see inset) by 1/10 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 50/log 10 = 1.6990. Images generated with Fractal Generator for ImageJFractal Generator for ImageJ .. 1.7227 Pinwheel fractal Built with Conway's Pinwheel tile. 1.7712 Sphinx fractal Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.W. Trump, G. Huber, C. Knecht, R. Ziff, to be published 1.7712 Hexaflake Built by exchanging iteratively each hexagon by a flake of 7 hexagons. Its boundary is the von Koch flake and contains an infinity of Koch snowflakes (black or white). 1.7712 Fractal H-I de Rivera Starting from a unit square dividing its dimensions into three equal parts to form nine self-similar squares with the first square, two middle squares (the one that is above and the one below the central square) are removed in each of the seven squares not eliminated the process is repeated, so it continues indefinitely. 1.7848 Von Koch curve 85° Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then . 1.8272 A self-affine fractal set Build iteratively from a p-by-q array on a square, with . Its Hausdorff dimension equals with and is the number of elements in the th column. The box-counting dimension yields a different formula, therefore, a different value. Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case. 1.8617 Pentaflake Built by exchanging iteratively each pentagon by a flake of 6 pentagons.