List of fractals by Hausdorff dimension

# List of fractals by Hausdorff dimension

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According to Falconer, one of the essential features of a fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

is that its Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...

strictly exceeds its topological dimension.
Presented here is a list of fractals ordered by increasing Hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension.

## Random and natural fractals

 Hausdorf dimension(exact value) Hausdorf dimension(approx.) Name Illustration Remarks 1/2 0.5 Zeros of a Wiener processWiener processIn mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown... The zeros of a Wiener process (Brownian motion) are a nowhere dense setNowhere dense setIn mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it... of Lebesgue measureLebesgue measureIn measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called... 0 with a fractal structure. Solution of where and 0.7499 a random Cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.... with 50% - 30% Generalization : At each iteration, the length of the left interval is defined with a random variable , a variable percentage of the length of the original interval. Same for the right interval, with a random variable . Its Hausdorff Dimension satisfies : . ( is the expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on... of ). Solution of 1.144... von Koch curve with random interval The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3). Measured 1.25 Coastline of Great BritainHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2... Fractal dimension of the west coast of Great Britain, as measured by Lewis Fry RichardsonLewis Fry RichardsonLewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them... and cited by Benoît MandelbrotBenoît MandelbrotBenoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child.... 1.2619 von Koch curve with random orientation One introduces here an element of randomness which does not affect the dimension, by choosing, at each iteration, to place the equilateral triangle above or below the curve. 1.333 Boundary of Brownian motion (cf. Mandelbrot, Lawler, SchrammOded SchrammOded Schramm was an Israeli-American mathematician known for the invention of the Schramm–Loewner evolution and for working at the intersection of conformal field theory and probability theory.-Biography:..., WernerWendelin WernerWendelin Werner is a German-born French mathematician working in the area of self-avoiding random walks, Schramm-Loewner evolution, and related theories in probability theory and mathematical physics. In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain he received the...). 1.333 2D polymer Similar to the brownian motion in 2D with non self-intersection. 1.333 Percolation front in 2D, Corrosion front in 2D Fractal dimension of the percolation-by-invasion front (accessible perimeter), at the percolation thresholdPercolation thresholdPercolation threshold is a mathematical term related to percolation theory, which is the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist while above it, there exists a giant component of the order of system size...  (59.3%). It’s also the fractal dimension of a stopped corrosion front. 1.40 Clusters of clusters 2DDiffusion-limited aggregationDiffusion-limited aggregation is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means... When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. 1.5 Graph of a regular Brownian function (Wiener processWiener processIn mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...) Graph of a function f such that, for any two positive reals x and x+h, the difference of their images has the centered gaussian distribution with variance = h. Generalization : The fractional Brownian motion of index follows the same definition but with a variance , in that case its Hausdorff dimension =. Measured 1.52 Coastline of NorwayFjordGeologically, a fjord is a long, narrow inlet with steep sides or cliffs, created in a valley carved by glacial activity.-Formation:A fjord is formed when a glacier cuts a U-shaped valley by abrasion of the surrounding bedrock. Glacial melting is accompanied by rebound of Earth's crust as the ice... See J. Feder. Measured 1.55 Random walk with no self-intersection Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends. 1.66 3D polymer Similar to the brownian motion in a cubic lattice, but without self-intersection. 1.70 2D DLA ClusterDiffusion-limited aggregationDiffusion-limited aggregation is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means... In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70. 1.7381 Fractal percolation with 75% probability The fractal percolation model is constructed by the progressive replacement of each square by a 3x3 grid in which is placed a random collection of sub-squares, each sub-square being retained with probability p. The "almost sure" Hausdorff dimension equals . 7/4 1.75 2D percolation cluster hull The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk, or by Schramm-Loewner Evolution. 1.8958 2D percolation cluster In a square lattice, under the site percolation thresholdPercolation thresholdPercolation threshold is a mathematical term related to percolation theory, which is the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist while above it, there exists a giant component of the order of system size... (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48. Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ». 2 Brownian motionBrownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications... Or random walk. The Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets"). Measured Around 2 Distribution of galaxy clusterGalaxy clusterA galaxy cluster is a compact cluster of galaxies. Basic difference between a galaxy group and a galaxy cluster is that there are many more galaxies in a cluster than in a group. Also, galaxies in a cluster are more compact and have higher velocity dispersion. One of the key features of cluster is...s From the 2005 results of the Sloan Digital Sky Survey. 2.33 CauliflowerCauliflowerCauliflower is one of several vegetables in the species Brassica oleracea, in the family Brassicaceae. It is an annual plant that reproduces by seed... Every branch carries around 13 branches 3 times smaller. 2.5 Balls of crumpled paper When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the ISO 216ISO 216ISO 216 specifies international standard paper sizes used in most countries in the world today. It defines the "A" and "B" series of paper sizes, including A4, the most commonly available size... A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made. Creases will form at all size scales (see Universality (dynamical systems)Universality (dynamical systems)In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...). 2.50 3D DLA ClusterDiffusion-limited aggregationDiffusion-limited aggregation is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means... In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50. 2.50 Lichtenberg figureLichtenberg figureLichtenberg figures are branching electric discharges that sometimes appear on the surface or the interior of insulating materials. They are named after the German physicist Georg Christoph Lichtenberg, who originally discovered and studied them... Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA. 2.5 regular BrownianBrownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications... surface A function , gives the height of a point such that, for two given positive increments and , then has a centered Gaussian distribution with variance = . Generalization : The fractional Brownian surface of index follows the same definition but with a variance = , in that case its Hausdorff dimension = . Measured 2.52 3D percolationPercolation theoryIn mathematics, percolation theory describes the behavior of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.-Introduction:... cluster In a cubic lattice, at the site percolation thresholdPercolation thresholdPercolation threshold is a mathematical term related to percolation theory, which is the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist while above it, there exists a giant component of the order of system size... (31.1%), the 3D percolation-by-invasion cluster has a fractal dimension of around 2.52. Beyond that threshold, the cluster is infinite. Measured 2.66 BroccoliBroccoliBroccoli is a plant in the cabbage family, whose large flower head is used as a vegetable.-General:The word broccoli, from the Italian plural of , refers to "the flowering top of a cabbage".... 2.79 Surface of human brainCerebral cortexThe cerebral cortex is a sheet of neural tissue that is outermost to the cerebrum of the mammalian brain. It plays a key role in memory, attention, perceptual awareness, thought, language, and consciousness. It is constituted of up to six horizontal layers, each of which has a different... 2.97 Lung surface The alveoli of a lung form a fractal surface close to 3. Calculated Multiplicative cascadeMultiplicative CascadeIn mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.Model I :Model II :Model III :... This is an example of a multifractal distribution. However by choosing its parameters in a particular way we can force the distribution to become a monofractal.

• Fractal dimension
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

• Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...

• Scale invariance
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...