Fractal analysis

Underlying principles

Fractals have fractional dimensions, which are a measure of complexity that indicates the degree to which the objects fill the available space. The fractal dimension measures the change in "size" of a fractal set with the changing observational scale, and is not limited by integer values. This is possible given that a smaller section of the fractal resembles the entirety, showing the same statistical properties at different scales. This characteristic is termed scale invariance, and can be further categorized as self-similarity or self-affinity, the latter scaled anisotropically (depending on the direction). Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex. Fractal analysis uses these underlying properties to help in the understanding and characterization of complex systems. It is also possible to expand the use of fractals to the lack of a single characteristic time scale, or pattern.

Further information on the Origins: Fractal Geometry

Types of fractal analysis

There are various types of fractal analysis, including box counting, lacunarity analysis, mass methods, and multifractal analysis. A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs. These can be acquired with various types of fractal generating software capable of generating benchmark patterns suitable for this purpose, which generally differ from software designed to render fractal art. Other types include detrended fluctuation analysis and the Hurst absolute value method, which estimate the hurst exponent.

Applications

Ecology and evolution

Unlike theoretical fractal curves which can be easily measured and the underlying mathematical properties calculated; natural systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial self-similarity. Using fractal analysis, it is possible to analyze and recognize when features of complex ecological systems are altered since fractals are able to characterize the natural complexity in such systems. Thus, fractal analysis can help to quantify patterns in nature and to identify deviations from these natural sequences. It helps to improve our overall understanding of ecosystems and to reveal some of the underlying structural mechanisms of nature. For example, it was found that the structure of an individual tree's xylem follows the same architecture as the spatial distribution of the trees in the forest, and that the distribution of the trees in the forest shared the same underlying fractal structure as the branches, scaling identically to the point of being able to use the pattern of the trees' branches mathematically to determine the structure of the forest stand. The use of fractal analysis for understanding structures, and spatial and temporal complexity in biological systems has already been well studied and its use continues to increase in ecological research. Despite its extensive use, it still receives some criticism.

Architecture, urban design and landscape design

In his publication The Fractal Geometry of Nature, Benoit Mandelbrot suggested fractal theory could be applied to architecture. In this context, Mandelbrot was talking about the self-similar feature of fractal objects, rather than fractal analysis. In 1996, Carl Bovill applied the box counting method of fractal analysis to Architecture. Bovill's work, using a manual version of box counting, has since been refined by others and computational approaches have been developed.

Fractal analysis is one of the few quantitative analysis methods available to architects and designers to understand the visual complexity of buildings, urban areas and landscapes. Typical uses of fractal analysis of the built environment have been to understand the visual complexity of cities and skylines, the fractal dimensions of works of different architects and the landscape.

Combining the fractal analysis of ecology (see above) with fractal analysis of architecture, fractal dimensions have been used to explore the possible relationship between nature and architecture.{{cite conference

Animal behaviour

Patterns in animal behaviour exhibit fractal properties on spatial and temporal scales. Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time. Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns. This has generated ecological interpretations such as the Lévy Flight Foraging hypothesis, which has proven to be a more accurate description of animal movement for some species.

Spatial patterns and animal behaviour sequences in fractal time have an optimal complexity range, which can be thought of as the homeostatic state on the spectrum where the complexity sequence should regularly fall. An increase or a loss in complexity, either becoming more stereotypical or conversely more random in their behaviour patterns, indicates that there has been an alteration in the functionality of the individual. Using fractal analysis, it is possible to examine the movement sequential complexity of animal behaviour and to determine whether individuals are experiencing deviations from their optimal range, suggesting a change in condition. For example, it has been used to assess welfare of domestic hens, stress in bottlenose dolphins in response to human disturbance, and parasitic infection in Japanese macaques and sheep. The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships. When it comes to animal welfare and conservation, fractal analysis makes it possible to identify potential sources of stress on animal behaviour, stressors that may not always be discernible through classical behaviour research.

This approach is more objective than classical behaviour measurements, such as frequency-based observations that are limited by the counts of behaviours, but is able to delve into the underlying reason for the behaviour. Another important advantage of fractal analysis is the ability to monitor the health of wild and free-ranging animal populations in their natural habitats without invasive measurements.

Applications include

Applications of fractal analysis include:

• Heart rate analysis{{cite journal |access-date=2025-11-04}}
• Human gait, balance, and activity
• Human anatomy
• Diagnostic imaging
• Cancer research
• Fractal analysis of complex networks
• Classification of histopathology slides in medicine |editor1-last = Losa |editor1-first= Gabriele A. |editor2-last= Nonnenmacher |editor2-first= Theo |access-date=1 February 2012
• Fractal landscape or Coastline complexity
• Electrical engineering
• Enzyme/enzymology (Michaelis-Menten kinetics)
• Generation of new music
• Generation of various art forms
• Search and rescue
• Signal and image compression
• Urban growth
• Neuroscience
• Diagnostic imaging{{Cite journal | doi-access = free
• Pathology{{Cite journal
• Geology
• Geography
• Archaeology{{Cite journal
• Seismology
• Soil studies
• Computer and video game design, especially computer graphics for organic environments and as part of procedural generation
• Fractography and fracture mechanics
• Fractal antennas — Small size antennas using fractal shapes
• Small angle scattering theory of fractally rough systems
• Generation of patterns for camouflage, such as MARPAT
• Digital sundial
• Technical analysis of price series (see Elliott wave principle)
• Fractal calculus

References

References

1. (2021). "Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning". Fractals.
2. Seuront, Laurent. (2009-10-12). "Fractals and Multifractals in Ecology and Aquatic Science". CRC Press.
3. Peters, Edgar. (1996). "Chaos and order in the capital markets: a new view of cycles, prices, and market volatility". Wiley.
4. Mulligan, R.. (2004). "Fractal analysis of highly volatile markets: an application to technology equities". The Quarterly Review of Economics and Finance.
5. Kamenshchikov, S.. (2014). "Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series". Journal of Chaos.
6. (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology.
7. (2014). "Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke". PLOS ONE.
8. (2008). "2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society".
9. Fractal Analysis of Digital Images [https://web.archive.org/web/20090131193936/http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm]
10. "Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society". Fractals: An Interdiscipinary Journal on the Complex Geometry of Nature.
11. Benoît B. Mandelbrot. (1983). "The fractal geometry of nature". Macmillan.
12. Khalili Golmankhaneh, Alireza. (2022). "Fractal Calculus and its Applications". World Scientific Pub Co Inc.
13. Mandelbrot, B.. (1967-05-05). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science.
14. (January 2002). "What is physiologic complexity and how does it change with aging and disease?". Neurobiology of Aging.
15. "Digital Images in FracLac". ImageJ.
16. (December 2013). "Temporal fractals in seabird foraging behaviour: diving through the scales of time". Scientific Reports.
17. Frontier, Serge. (1987). "Developments in Numerical Ecology". Springer Berlin Heidelberg.
18. (August 1994). "Application of multifractals to the analysis of vegetation pattern". Journal of Vegetation Science.
19. (1998). "Spatio-temporal structure of tidally mixed coastal waters: variability and heterogeneity". Journal of Plankton Research.
20. (September 2003). "Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens". Applied Animal Behaviour Science.
21. (1983). "Fractal Dimension of a Coral Reef at Ecological Scales". Marine Ecology Progress Series.
22. (January 1982). "Time scales, persistence and patchiness". Biosystems.
23. West, G. B.. (1997-04-04). "A General Model for the Origin of Allometric Scaling Laws in Biology". Science.
24. (2009-04-28). "A general quantitative theory of forest structure and dynamics". Proceedings of the National Academy of Sciences.
25. (1991). "Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications". Soil Science Society of America Journal.
26. (April 1985). "Fractal dimension of vegetation and the distribution of arthropod body lengths". Nature.
27. (January 1998). "Fractal dimensions of small (15–200 μm) particles in Eastern Pacific coastal waters". Deep Sea Research Part I: Oceanographic Research Papers.
28. (May 2006). "Multifractals, cloud radiances and rain". Journal of Hydrology.
29. (2004-02-24). "Uses and abuses of fractal methodology in ecology". Ecology Letters.
30. (December 2012). "Revisiting detrended fluctuation analysis". Scientific Reports.
31. (1982). "The Fractal Geometry of Nature". Freeman.
32. (1996). "Fractal Geometry in Architecture and Design". Birkhauser.
33. (2010). "Proceedings of the 15th Conference on Computer Aided Architectural Design Research in Asia (CAADRIA)".
34. (2009). "Proceedings of the 27th International Conference on Education and Research in Computer Aided Architectural Design in Europe (ECAADe)".
35. (2000). "The fractal dimension of Tokyo's streets". Fractals.
36. (2015). "Characteristic visual complexity: Fractal dimensions in the architecture of frank lloyd wright and le corbusier. In:Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future". Springer International Publishing.
37. (2020). "Fractal geometry for landscape architecture: Review of methodologies and interpretations". Journal of Digital Landscape Architecture.
38. (2022). "Measuring the geometry of nature and architecture: comparing the visual properties of Frank Lloyd Wright's Fallingwater and its natural setting". Open House International.
39. (2003-10-28). "Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton". Proceedings of the National Academy of Sciences.
40. (September 2005). "Characterisation by fractal analysis of foraging paths of ewes grazing heterogeneous swards". Applied Animal Behaviour Science.
41. (2012-05-08). "Foraging success of biological Levy flights recorded in situ". Proceedings of the National Academy of Sciences.
42. (2009-10-30). "Lévy flights and random searches". Journal of Physics A: Mathematical and Theoretical.
43. (June 2001). "Lévy flights search patterns of biological organisms". Physica A: Statistical Mechanics and Its Applications.
44. MacIntosh, Andrew James Jonathan. (2014). "The Fractal Primate". Primate Research.
45. (August 2018). "Fractal measures in activity patterns: Do gastrointestinal parasites affect the complexity of sheep behaviour?". Applied Animal Behaviour Science.
46. (2011-10-07). "Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease". Journal of the Royal Society Interface.
47. (September 2016). "Changes in the behavioural complexity of bottlenose dolphins along a gradient of anthropogenically-impacted environments in South Australian coastal waters: Implications for conservation and management strategies". Journal of Experimental Marine Biology and Ecology.
48. (2014-05-01). "Complexity and behavioral ecology". Behavioral Ecology.
49. (February 1996). "Fractal structure of sequential behaviour patterns: an indicator of stress". Animal Behaviour.
50. (February 2004). "Fractal analysis of animal behaviour as an indicator of animal welfare". Animal Welfare.
51. "Applications".
52. (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology.
53. (2017). "Altered Functional Performance in Patients with Fibromyalgia". Frontiers in Human Neuroscience.
54. (2024). "Modern Methods of Neuroanatomical and Neurophysiological Research". MethodsX.
55. (2009). "Nest expansion assay: A cancer systems biology approach to in vitro invasion measurements". BMC Research Notes.
56. (25 November 2021). "Deciphering the generating rules and functionalities of complex networks". Scientific Reports.
57. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science.
58. (2008). "Proteins: Coexistence of Stability and Flexibility". Physical Review Letters.
59. Panteha Saeedi, and Soren A. Sorensen. (2009). "Proceedings of the World Congress on Engineering 2009". Newswood Limited.
60. (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE.
61. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder". Fractals.
62. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal.
63. (2008). "Properties of multivariate data investigated by fractal dimensionality". Journal of Neuroscience Methods.
64. Al-Kadi O.S, Watson D.. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images". IEEE Transactions on Biomedical Engineering.
65. (2011). "Fractals in microscopy". Journal of Microscopy.
66. (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology.
67. (2005). "The Broken Past: Fractals in Archaeology". Journal of Archaeological Method and Theory.
68. (2007). "Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing". Earth and Planetary Science Letters.
69. (2012). "Multifractal characterization of urban residential land price in space and time". Applied Geography.