List of Fractals By Hausdorff 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 function for the critical parameter value, where the period doubling is infinite. Notice that 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.
	0.6942	Asymmetric Cantor set		Note that the dimension is not, as would be expected from the generalized Cantor set with γ=1/4, which has the same length at each stage. Built by removing the second quarter at each iteration. Nowhere dense and not a countable set. (golden ratio).
	0.69897	Real numbers with even digits		Similar to a Cantor set.
	0.88137	Spectrum of Fibonacci Hamiltonian		The study 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.
	0	Generalized Cantor set		Built by removing at the th iteration the central interval of length from each remaining segment (of length ). At one obtains the usual Cantor set. Varying between 0 and 1 yields any fractal dimension .
	1	Smith–Volterra–Cantor set		Built by removing a central interval of length of each remaining interval at the nth iteration. Nowhere dense but has a Lebesgue measure of ½.
	1	Takagi or Blancmange curve		Defined on the unit interval by, where is the sawtooth function. Special case of the Takahi-Landsberg curve: with . The Hausdorff dimension equals for in . (Hunt cited by Mandelbrot).
Calculated	1.0812	Julia set z² + 1/4		Julia set for c = 1/4.
Solution s of	1.0933	Boundary of the Rauzy fractal		Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism:, and . 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 for parameters: Real = 0 and Imaginary = 1.
	1.2083	Fibonacci word fractal 60°		Build 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).
	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	Koch curve		3 von 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. Note that generating the pattern using statistical instead of exact self-similarity yields the same fractal dimension.
Calculated	1.2683	Julia set z2 − 1		Julia set for c = −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
	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
Calculated	1.3934	Douady rabbit		Julia set for c = −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).
	1.49	Quadric cross
(conjectured exact)	1.5000	a Weierstrass function:		The Hausdorff dimension of the Weierstrass function defined by with and has upper bound . It is believed to be the exact value. The same result can be established when, instead of the sine function, we use other periodic functions, like cosine.
	1.5000	Quadratic von Koch curve (type 2)		Also called "Minkowski sausage".
$\textstyle{\frac{\log\left(\frac{1+\sqrt{73-6\sqrt{87}}+\sqrt{73+6\sqrt{87}}}{3}\right)} {\log(2)}}$	1.5236	Boundary of the Dragon curve		cf. Chang & Zhang.
$\textstyle{\frac{\log\left(\frac{1+\sqrt{73-6\sqrt{87}}+\sqrt{73+6\sqrt{87}}}{3}\right)} {\log(2)}}$	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).
	1.5849	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.
	1.5849	Sierpinski triangle		Also the triangle of Pascal modulo 2.
	1.5849	Sierpiński arrowhead curve		Same limit as the triangle (above) but built with a one-dimensional curve.
	1.5849	Boundary of the T-Square fractal
	1.61803	a golden dragon		Built from two similarities of ratios and, with . Its dimension equals because . With (Golden number).
	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. Notice the presence of the Koch snowflake 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). (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 : .
	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.
	1.7	50 segment quadric fractal		Built with ImageJ
	1.7227	Pinwheel fractal		Built with Conway's Pinwheel tile.
	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).
	1.7848	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 array on a square, with . Its Hausdorff dimension equals with and is the number of elements in the 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. (golden ratio).
solution of	1.8687	Monkeys tree		This curve appeared in Benoit Mandelbrot's "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio and 5 similarities of ratio .
	1.8928	Sierpinski carpet		Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
	1.8928	3D Cantor dust		Cantor set in 3 dimensions.
	1.8928	Cartesian product of the von Koch curve and the Cantor set		Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then . See also the 2D Cantor dust and the Cantor cube.
Estimated	1.9340	Boundary of the Lévy C curve		Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
	1.974	Penrose tiling		See Ramachandrarao, Sinha & Sanyal.
	2	Boundary of the Mandelbrot set		The boundary and the set itself have the same dimension.
	2	Julia set		For determined values of c (including c belonging to the boundary of the Mandelbrot set), the Julia set has a dimension of 2.
	2	Sierpiński curve		Every Peano curve filling the plane has a Hausdorff dimension of 2.
	2	Hilbert curve
	2	Peano curve		And a family of curves built in a similar way, such as the Wunderlich curves.
	2	Moore curve		Can be extended in 3 dimensions.
	2	Lebesgue curve or z-order curve		Unlike the previous ones this space-filling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the Hilbert Curve it can be extended in 3D.
	2	Dragon curve		And its boundary has a fractal dimension of 1.5236270862.
	2	Terdragon curve		L-system: F → F + F – F, angle = 120°.
	2	Gosper curve		Its boundary is the Gosper island.
Solution of	2	Curve filling the Koch snowflake		Proposed by Mandelbrot in 1982, it fills the Koch snowflake. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio .
	2	Sierpiński tetrahedron		Each tetrahedron is replaced by 4 tetrahedra.
	2	H-fractal		Also the Mandelbrot tree which has a similar pattern.
	2	Pythagoras tree (fractal)		Every square generates two squares with a reduction ratio of sqrt(2)/2.
	2	2D Greek cross fractal		Each segment is replaced by a cross formed by 4 segments.
Measured	2.01 ±0.01	Rössler attractor		The fractal dimension of the Rössler attractor is slightly above 2. For a=0.1, b=0.1 and c=14 it has been estimated between 2.01 and 2.02.
Measured	2.06 ±0.01	Lorenz attractor		For parameters v=40,=16 and b=4 . See McGuinness (1983)
	2.3219	Fractal pyramid		Each pyramid is replaced by 5 pyramids, twice smaller. Must not be confused with the Sierpinski tetrahedron, since it is based on a square pyramid.
	2.3296	Dodecahedron fractal		Each dodecahedron is replaced by 20 dodecahedra. (golden ratio).
	2.3347	3D quadratic Koch surface (type 1)		Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
	2.4739	Apollonian sphere packing		The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.
	2.50	3D quadratic Koch surface (type 2)		Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
	2.5237	Cantor tesseract	no image available	Cantor set in 4 dimensions. Generalization: in a space of dimension n, the Cantor set has a Hausdorff dimension of .
	2.5819	Icosahedron fractal		Each icosahedron is replaced by 12 icosahedra. (golden ratio).
	2.5849	3D Greek cross fractal		Each segment is replaced by a cross formed by 6 segments.
	2.5849	Octahedron fractal		Each octahedron is replaced by 6 octahedra.
	2.5849	von Koch surface		Each equilateral triangular face is cut into 4 equal triangles. Using the central triangle as the base, form a tetrahedron. Replace the triangular base with the tetrahedral "tent".
	2.7268	Menger sponge		And its surface has a fractal dimension of, which is the same as that by volume.
	3	3D Hilbert curve		A Hilbert curve extended to 3 dimensions.
	3	3D Lebesgue curve		A Lebesgue curve extended to 3 dimensions.
	3	3D Moore curve		A Moore curve extended to 3 dimensions.
	3	3D H-fractal		A H-fractal extended to 3 dimensions.
	3	Mandelbulb		Extension of the Mandelbrot set (power 8) in 3 dimensions

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List of Fractals By Hausdorff Dimension - Deterministic Fractals