A fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.
The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale.
One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but rather is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which in this case is a number between one and two.
A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale.:1 Several types of fractal dimension can be measured theoretically and empirically (see Fig. 2). Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract to practical phenomena, including turbulence,:97–104 river networks:246–247, urban growth, human physiology, medicine, and market trends. The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s,:19 but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975.
Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points (0-dimensional sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3 for sets describing volumes (3-dimensional sets having length, width, and height). But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry.
Unlike topological dimensions, the fractal index can take non-integer values, indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. For instance, a curve with fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.:48[notes 1] This general relationship can be seen in the two images of fractal curves in Fig.2 and Fig. 3 – the 32-segment contour in Fig. 2, convoluted and space filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of 1.26.
The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated. Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity.[notes 2] These features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their length or slope, as with ordinary lines. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary lines lack. The self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. The length between any two points on these curves is undefined because the curves are theoretical constructs that never stop repeating themselves. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. These are not rectifiable curves, meaning they cannot be measured by being broken down into many segments approximating their respective lengths. They cannot be characterized by finding their lengths or slopes. However, their fractal dimensions can be determined, which shows that both fill space more than ordinary lines but less than surfaces, and allows them to be compared in this regard.
Note that the two fractal curves described above show a type of self-similarity that is exact with a repeating unit of detail that is readily visualized. This sort of structure can be extended to other spaces (e.g., a fractal that extends the Koch curve into 3-d space has a theoretical D=2.5849). However, such neatly countable complexity is only one example of the self-similarity and detail that are present in fractals. The example of the coast line of Britain, for instance, exhibits self-similarity of an approximate pattern with approximate scaling.:26 Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. These include, as examples, strange attractors for which the detail has been described as in essence, smooth portions piling up,:49 the Julia set, which can be seen to be complex swirls upon swirls, and heart rates, which are patterns of rough spikes repeated and scaled in time. Fractal complexity may not always be resolvable into easily grasped units of detail and scale without complex analytic methods but it is still quantifiable through fractal dimensions.:197; 262
The terms fractal dimension and fractal were coined by Mandelbrot in 1975, about a decade after he published his paper on self-similarity in the coastline of Britain. Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in usual linear terms. The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about undifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s.:405 There was a lull in the published work on such functions for a time after that, then a renewal starting in the late 1800s with the publishing of mathematical functions and sets that are today called canonical fractals (such as the eponymous works of von Koch, Sierpiński, and Julia), but at the time of their formulation were often considered antithetical mathematical "monsters". These works were accompanied by perhaps the most pivotal point in the development of the concept of a fractal dimension through the work of Hausdorff in the early 1900s who defined a "fractional" dimension that has come to be named after him and is frequently invoked in defining modern fractals.:44
See Fractal history for more information
Role of scaling
The concept of a fractal dimension rests in unconventional views of scaling and dimension. As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable stands for the number of new sticks, for the scaling factor, and for the fractal dimension:
The symbol above denotes proportionality. This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because =3 when =1/3 as in the example above, =1, and for squares, because =9 when =1/3, =2.
The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, =4 when =1/3, and the value of can be found by rearranging Equation 1:
That is, for a fractal described by =4 when =1/3, =1.2619, a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in. The scaling used in this example is the same scaling of the Koch curve and snowflake. Of note, these images themselves are not true fractals because the scaling described by the value of cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619.
D is not a unique descriptor
As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns. The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Many fractal structures or patterns could be constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Figure 6.
The concept of fractal dimension described in this article is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practice, however, fractal dimensions can be determined using techniques that approximate scaling and detail from limits estimated from regression lines over log vs log plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for some classic fractals all these dimensions coincide, in general they are not equivalent:
- Information dimension: D considers how the average information needed to identify an occupied box scales with box size; is a probability.
- Correlation dimension D is based on as the number of points used to generate a representation of a fractal and gε, the number of pairs of points closer than ε to each other.
- Generalized or Rényi dimensions
- The box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of generalized dimensions of order α, defined by:
- Multifractal dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern.
- Uncertainty exponent
- Hausdorff dimension
- Packing dimension
- Local connected dimension
Estimating from real-world data
Many real-world phenomena exhibit limited or statistical fractal properties and fractal dimensions that have been estimated from sampled data using computer based fractal analysis techniques. Practically, measurements of fractal dimension are affected by various methodological issues, and are sensitive to numerical or experimental noise and limitations in the amount of data. Nonetheless, the field is rapidly growing as estimated fractal dimensions for statistically self-similar phenomena may have many practical applications in various fields including diagnostic imaging, physiology, neuroscience, medicine, physics, image analysis,  acoustics, Riemann zeta zeros, and electrochemical processes.
An alternative to a direct measurement, is considering a mathematical model that resembles formation of a real-world fractal object. In this case, a validation can also be done by comparing other than fractal properties implied by the model, with measured data. In colloidal physics, systems composed of particles with various fractal dimensions arise. To describe these systems, it is convenient to speak about a distribution of fractal dimensions, and eventually, a time evolution of the latter: a process that is driven by a complex interplay between aggregation and coalescence. 
- Falconer, Kenneth (2003). Fractal Geometry. New York: Wiley. p. 308. ISBN 978-0-470-84862-3.
- Sagan, Hans (1994). Space-Filling Curves. Berlin: Springer-Verlag. p. 156. ISBN 0-387-94265-3.
- Vicsek, Tamás (1992). Fractal growth phenomena. Singapore New Jersey: World Scientific. p. 10. ISBN 978-981-02-0668-0.
- Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science 156 (3775): 636–638. doi:10.1126/science.156.3775.636. PMID 17837158.
- Benoit B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012.
- Harte, David (2001). Multifractals. London: Chapman & Hall. pp. 3–4. ISBN 978-1-58488-154-4.
- Karperien (2004). Defining Microglial Morphology: Form, Function, and Fractal Dimension. Charles Sturt University. p. 86. Retrieved 9 July 2013.
- Losa, Gabriele A.; Nonnenmacher, Theo F., eds. (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2. Retrieved 1 February 2012.
- Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLoS ONE 6 (9): e24791. doi:10.1371/journal.pone.0024791. PMC 3176775. PMID 21949753.
- "Applications". Retrieved 2007-10-21.
- Popescu, D. P.; Flueraru, C.; Mao, Y.; Chang, S.; Sowa, M. G. (2010). "Signal attenuation and box-counting fractal analysis of optical coherence tomography images of arterial tissue". Biomedical Optics Express 1 (1): 268–277. doi:10.1364/boe.1.000268. PMC 3005165. PMID 21258464.
- King, R. D.; George, A. T.; Jeon, T.; Hynan, L. S.; Youn, T. S.; Kennedy, D. N.; Dickerson, B.; the Alzheimer’s Disease Neuroimaging Initiative (2009). "Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis". Brain Imaging and Behavior 3 (2): 154–166. doi:10.1007/s11682-008-9057-9. PMC 2927230. PMID 20740072.
- Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. New York: Wiley. ISBN 0-471-13938-6.
- Edgar, Gerald (2004). Classics on Fractals. Boulder: Westview Press. ISBN 978-0-8133-4153-8.
- Albers; Alexanderson (2008). "Benoit Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, Mass: AK Peters. p. 214. ISBN 978-1-56881-340-0.
- Mandelbrot, Benoit (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN 978-0-387-20158-0. "A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension"
- Sharifi-Viand, A.; Mahjani, M. G.; Jafarian, M. (2012). "Investigation of anomalous diffusion and multifractal dimensions in polypyrrole film". Journal of Electroanalytical Chemistry 671: 51. doi:10.1016/j.jelechem.2012.02.014.
- Helge von Koch, "On a continuous curve without tangents constructible from elementary geometry" In Gerald Edgar, ed. (2004). Classics on Fractals. Boulder: Westview Press. pp. 25–46. ISBN 978-0-8133-4153-8.
- Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology 587 (15): 3929. doi:10.1113/jphysiol.2009.169219.
- Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7.
- Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on 4 February 2012.
- Iannaccone, Khokha (1996). Fractal Geometry in Biological Systems. ISBN 978-0-8493-7636-8.
- Vicsek, Tamás (2001). Fluctuations and scaling in biology. Oxford [Oxfordshire]: Oxford University Press. ISBN 0-19-850790-9.
- Jelinek, A.; Jelinek, H. F.; Leandro, J. J.; Soares, J. V.; Cesar Jr, R. M.; Luckie, A. (2008). "Automated detection of proliferative retinopathy in clinical practice". Clinical Ophthalmology 2 (1): 109–122. doi:10.2147/OPTH.S1579. PMC 2698675. PMID 19668394.
- Landini, G.; Murray, P. I.; Misson, G. P. (1995). "Local connected fractal dimensions and lacunarity analyses of 60 degrees fluorescein angiograms". Investigative Ophthalmology & Visual Science 36 (13): 2749–2755. PMID 7499097.
- Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology 29 (7): 919–932. doi:10.1023/A:1022355723781.
- Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal 85 (6): 4041–4046. doi:10.1016/S0006-3495(03)74817-6. PMC 1303704. PMID 14645092.
- Smith, T. G.; Lange, G. D.; Marks, W. B. (1996). "Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals". Journal of Neuroscience Methods 69 (2): 123–136. doi:10.1016/S0165-0270(96)00080-5. PMID 8946315.
- Li, J.; Du, Q.; Sun, C. (2009). "An improved box-counting method for image fractal dimension estimation". Pattern Recognition 42 (11): 2460. doi:10.1016/j.patcog.2009.03.001.
- Dubuc, B.; Quiniou, J.; Roques-Carmes, C.; Tricot, C.; Zucker, S. (1989). "Evaluating the fractal dimension of profiles". Physical Review A 39 (3): 1500–1512. Bibcode:1989PhRvA..39.1500D. doi:10.1103/PhysRevA.39.1500. PMID 9901387.
- Roberts, A.; Cronin, A. (1996). "Unbiased estimation of multi-fractal dimensions of finite data sets". Physica A: Statistical Mechanics and its Applications 233 (3–4): 867. doi:10.1016/S0378-4371(96)00165-3.
- Al-Kadi O.S, Watson D. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images". IEEE Transactions on Biomedical Engineering 55 (7): 1822–1830. doi:10.1109/tbme.2008.919735.
- Pierre Soille and Jean-F. Rivest (1996). "On the Validity of Fractal Dimension Measurements in Image Analysis". Journal of Visual Communication and Image Representation 7 (3): 217–229. doi:10.1006/jvci.1996.0020. ISSN 1047-3203.
- Tolle, C. R.; McJunkin, T. R.; Gorsich, D. J. (2003). "Suboptimal minimum cluster volume cover-based method for measuring fractal dimension". IEEE Transactions on Pattern Analysis and Machine Intelligence 25: 32. doi:10.1109/TPAMI.2003.1159944.
- Gorsich, D. J.; Tolle, C. R.; Karlsen, R. E.; Gerhart, G. R. (1996). "Wavelet and fractal analysis of ground-vehicle images". Wavelet Applications in Signal and Image Processing IV. Wavelet Applications in Signal and Image Processing IV 2825. p. 109. doi:10.1117/12.255224.
- Maragos, P.; Potamianos, A. (1999). "Fractal dimensions of speech sounds: Computation and application to automatic speech recognition". The Journal of the Acoustical Society of America 105 (3): 1925–1932. doi:10.1121/1.426738. PMID 10089613.
- Shanker, O. (2006). "Random matrices, generalized zeta functions and self-similarity of zero distributions". Journal of Physics A: Mathematical and General 39 (45): 13983. doi:10.1088/0305-4470/39/45/008.
- Eftekhari, A. (2004). "Fractal Dimension of Electrochemical Reactions". Journal of the Electrochemical Society 151 (9): E291–E296. doi:10.1149/1.1773583.
- Kryven, I.; Lazzari, S.; Storti, G. (2014). "Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems". Macromolecular Theory and Simulations 23 (3): 170. doi:10.1002/mats.201300140.
- Mandelbrot, Benoit B., The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward (Basic Books, 2004)
- TruSoft's Benoit, fractal analysis software product calculates fractal dimensions and hurst exponents.
- A Java Applet to Compute Fractal Dimensions
- Introduction to Fractal Analysis
- Bowley, Roger (2009). "Fractal Dimension". Sixty Symbols. Brady Haran for the University of Nottingham.