Skip to content

References

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Foundational Works in Fractal Geometry

Mandelbrot, Benoit B. The Fractal Geometry of Nature. W.H. Freeman, 1982.

Falconer, Kenneth. Fractal Geometry: Mathematical Foundations and Applications. 3rd ed., Wiley, 2014.

Feder, Jens. Fractals. Plenum Press, 1988.

Diffusion Limited Aggregation

Witten, Thomas A., and Leonard M. Sander. "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon." Physical Review Letters, vol. 47, no. 19, 1981, pp. 1400--1403.

Witten, Thomas A., and Leonard M. Sander. "Diffusion-Limited Aggregation." Physical Review B, vol. 27, no. 9, 1983, pp. 5686--5697.

Meakin, Paul. "Diffusion-Controlled Cluster Formation in 2--6-Dimensional Space." Physical Review A, vol. 27, no. 3, 1983, pp. 1495--1507.

Meakin, Paul. Fractals, Scaling and Growth Far from Equilibrium. Cambridge UP, 1998.

Halsey, Thomas C. "Diffusion-Limited Aggregation: A Model for Pattern Formation." Physics Today, vol. 53, no. 11, 2000, pp. 36--41.

Sand Box Method and Dimension Estimation

Tél, Tamás, et al. "Determination of Fractal Dimensions for Geometrical Multifractals." Physica A, vol. 159, no. 2, 1989, pp. 155--166.

Voss, Richard F. "Characterization and Measurement of Random Fractals." Physica Scripta, vol. T13, 1986, pp. 27--32.

Lichen and Algal Morphology

Honegger, Rosmarie. "The Lichen Symbiosis---What Is So Spectacular About It?" The Lichenologist, vol. 30, no. 3, 1998, pp. 193--212.

Prigogine, Ilya, and Isabelle Stengers. Order Out of Chaos: Man's New Dialogue with Nature. Bantam Books, 1984.

Armstrong, Richard A. "Growth and Regeneration of Lichen Thalli with the Emphasis on the Central/Marginal Zonation Phenomenon." Symbiosis, vol. 29, 2000, pp. 7--15.

Metabolic Scaling Theory

West, Geoffrey B., James H. Brown, and Brian J. Enquist. "A General Model for the Origin of Allometric Scaling Laws in Biology." Science, vol. 276, no. 5309, 1997, pp. 122--126.

West, Geoffrey B., James H. Brown, and Brian J. Enquist. "A General Model for the Structure and Allometry of Plant Vascular Systems." Nature, vol. 400, no. 6745, 1999, pp. 664--667.

Brown, James H., et al. "Toward a Metabolic Theory of Ecology." Ecology, vol. 85, no. 7, 2004, pp. 1771--1789.

Enquist, Brian J. "Universal Scaling in Tree and Vascular Plant Allometry: Toward a General Quantitative Theory Linking Plant Form and Function from Cells to Ecosystems." Tree Physiology, vol. 22, no. 15--16, 2002, pp. 1045--1064.

Leonardo's Rule and Area-Preserving Branching

Richter, Jean Paul. The Notebooks of Leonardo da Vinci. Dover, 1970.

Eloy, Christophe. "Leonardo's Rule, Self-Similarity, and Wind-Induced Stresses in Trees." Physical Review Letters, vol. 107, no. 25, 2011, 258101.

Bentley, Lawrence P., et al. "An Empirical Assessment of Tree Branching Networks and Implications for Plant Allometric Scaling Models." Ecology Letters, vol. 16, no. 8, 2013, pp. 1069--1078.

Multifractal Analysis

Chhabra, Ashvin, and Roderick V. Jensen. "Direct Determination of the f(alpha) Singularity Spectrum." Physical Review Letters, vol. 62, no. 12, 1989, pp. 1327--1330.

Muzy, Jean-François, Emmanuel Bacry, and Alain Arneodo. "The Multifractal Formalism Revisited with Wavelets." International Journal of Bifurcation and Chaos, vol. 4, no. 2, 1994, pp. 245--302.

Arneodo, Alain, et al. "The Thermodynamics of Fractals Revisited with Wavelets." Physica A, vol. 213, no. 1--2, 1995, pp. 232--275.

Terrestrial Laser Scanning and Tree Architecture

Raumonen, Pasi, et al. "Fast Automatic Precision Tree Models from Terrestrial Laser Scanner Data." Remote Sensing, vol. 5, no. 2, 2013, pp. 491--520.

Hackenberg, Jan, et al. "SimpleTree---An Efficient Open Source Tool to Build Tree Models from TLS Clouds." Forests, vol. 6, no. 11, 2015, pp. 4245--4294.

Calders, Kim, et al. "Nondestructive Estimates of Above-Ground Biomass Using Terrestrial Laser Scanning." Methods in Ecology and Evolution, vol. 6, no. 2, 2015, pp. 198--208.

Lau, Alvaro, et al. "Quantifying Branch Architecture of Tropical Trees Using Terrestrial LiDAR and 3D Modelling." Trees, vol. 32, no. 5, 2018, pp. 1219--1231.

Self-Affine Fractals and Surface Analysis

Mandelbrot, Benoit B. "Self-Affine Fractals and Fractal Dimension." Physica Scripta, vol. 32, no. 4, 1985, pp. 257--260.

Sarkar, Nirupam, and B.B. Chaudhuri. "An Efficient Differential Box-Counting Approach to Compute Fractal Dimension of Image." IEEE Transactions on Systems, Man, and Cybernetics, vol. 24, no. 1, 1994, pp. 115--120.

Keller, James M., et al. "Texture Description and Segmentation through Fractal Geometry." Computer Vision, Graphics, and Image Processing, vol. 45, no. 2, 1989, pp. 150--166.

Canopy Structure and LiDAR Remote Sensing

Lefsky, Michael A., et al. "Lidar Remote Sensing for Ecosystem Studies." BioScience, vol. 52, no. 1, 2002, pp. 19--30.

Parker, Geoffrey G., and Michael E. Brown. "Forest Canopy Stratification---Is It Useful?" The American Naturalist, vol. 155, no. 4, 2000, pp. 473--484.

Zimble, David A., et al. "Characterizing Vertical Forest Structure Using Small-Footprint Airborne LiDAR." Remote Sensing of Environment, vol. 87, no. 2--3, 2003, pp. 171--182.

Asner, Gregory P., et al. "A Universal Airborne LiDAR Approach for Tropical Forest Carbon Mapping." Oecologia, vol. 168, no. 4, 2012, pp. 1147--1160.

Gap Dynamics and Self-Organized Criticality

Bak, Per, Chao Tang, and Kurt Wiesenfeld. "Self-Organized Criticality: An Explanation of 1/f Noise." Physical Review Letters, vol. 59, no. 4, 1987, pp. 381--384.

Bak, Per. How Nature Works: The Science of Self-Organized Criticality. Copernicus, 1996.

Solé, Ricard V., and Susanna C. Manrubia. "Are Rainforests Self-Organized in a Critical State?" Journal of Theoretical Biology, vol. 173, no. 1, 1995, pp. 31--40.

Pascual, Mercedes, and Frederic Guichard. "Criticality and Disturbance in Spatial Ecological Systems." Trends in Ecology and Evolution, vol. 20, no. 2, 2005, pp. 88--95.

Zeta Distribution and Power Laws

Zipf, George Kingsley. Human Behavior and the Principle of Least Effort. Addison-Wesley, 1949.

Newman, Mark E.J. "Power Laws, Pareto Distributions and Zipf's Law." Contemporary Physics, vol. 46, no. 5, 2005, pp. 323--351.

Clauset, Aaron, Cosma Rohilla Shalizi, and M.E.J. Newman. "Power-Law Distributions in Empirical Data." SIAM Review, vol. 51, no. 4, 2009, pp. 661--703.

Apollonian Gaskets

Mandelbrot, Benoit B. "The Fractal Geometry of Nature." Chapter 18: Self-Inverse Fractals, Apollonian Gaskets. W.H. Freeman, 1982, pp. 166--179.

Boyd, David W. "The Residual Set Dimension of the Apollonian Packing." Mathematika, vol. 20, no. 2, 1973, pp. 170--174.

Graham, Ronald L., et al. "Apollonian Circle Packings: Number Theory." Journal of Number Theory, vol. 100, no. 1, 2003, pp. 1--45.

Ecological Applications of Fractal Geometry

Hastings, Harold M., and George Sugihara. Fractals: A User's Guide for the Natural Sciences. Oxford UP, 1993.

Sugihara, George, and Robert M. May. "Applications of Fractals in Ecology." Trends in Ecology and Evolution, vol. 5, no. 3, 1990, pp. 79--86.

Halley, John M., et al. "Uses and Abuses of Fractal Methodology in Ecology." Ecology Letters, vol. 7, no. 3, 2004, pp. 254--271.

Frontier, Serge. "Applications of Fractal Theory to Ecology." Developments in Numerical Ecology, edited by Pierre Legendre and Louis Legendre, Springer, 1987, pp. 335--378.

Plant Hydraulics and Xylem Architecture

Tyree, Melvin T., and Martin H. Zimmermann. Xylem Structure and the Ascent of Sap. 2nd ed., Springer, 2002.

Sperry, John S., et al. "A Species-Level Model for Metabolic Scaling in Trees I. Exploring Boundaries to Scaling Space within and across Species." Functional Ecology, vol. 26, no. 5, 2012, pp. 1054--1065.

Olson, Mark E., et al. "Universal Hydraulics of the Flowering Plants: Vessel Diameter Scales with Stem Length across Angiosperm Lineages, Habits and Climates." Ecology Letters, vol. 17, no. 8, 2014, pp. 988--997.

Statistical Methods

Efron, Bradley, and Robert J. Tibshirani. An Introduction to the Bootstrap. Chapman and Hall, 1993.

Burnham, Kenneth P., and David R. Anderson. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. 2nd ed., Springer, 2002.

Cohen, Jacob. Statistical Power Analysis for the Behavioral Sciences. 2nd ed., Lawrence Erlbaum, 1988.

Thermodynamics and Biological Organization

Nicolis, Gregoire, and Ilya Prigogine. Self-Organization in Nonequilibrium Systems. Wiley, 1977.

Odum, Howard T., and Richard C. Pinkerton. "Time's Speed Regulator: The Optimum Efficiency for Maximum Power Output in Physical and Biological Systems." American Scientist, vol. 43, no. 2, 1955, pp. 331--343.

Schneider, Eric D., and James J. Kay. "Life as a Manifestation of the Second Law of Thermodynamics." Mathematical and Computer Modelling, vol. 19, no. 6--8, 1994, pp. 25--48.

Martyushev, Leonid M., and Vladimir D. Seleznev. "Maximum Entropy Production Principle in Physics, Chemistry and Biology." Physics Reports, vol. 426, no. 1, 2006, pp. 1--45.

Riemann Zeta Function and Fractal Strings

Lapidus, Michel L., and Machiel van Frankenhuijsen. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. 2nd ed., Springer, 2013.

Lapidus, Michel L., and Carl Pomerance. "The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums." Proceedings of the London Mathematical Society, vol. 66, no. 1, 1993, pp. 41--69.

Lapidus, Michel L. "Fractal Strings, Complex Dimensions and the Spectral Operator: From the Riemann Hypothesis to Phase Transitions and Universality." Journal of Physics A: Mathematical and Theoretical, vol. 52, 2019.

Random Matrix Theory and GUE Statistics

Mehta, Madan Lal. Random Matrices. 3rd ed., Academic Press, 2004.

Montgomery, Hugh L. "The Pair Correlation of Zeros of the Zeta Function." Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, vol. 24, 1973, pp. 181--193.

Odlyzko, Andrew M. "On the Distribution of Spacings Between Zeros of the Zeta Function." Mathematics of Computation, vol. 48, no. 177, 1987, pp. 273--308.

Berry, Michael V., and Jonathan P. Keating. "The Riemann Zeros and Eigenvalue Asymptotics." SIAM Review, vol. 41, no. 2, 1999, pp. 236--266.

Spectral Geometry and Complex Dimensions

Kac, Mark. "Can One Hear the Shape of a Drum?" The American Mathematical Monthly, vol. 73, no. 4, 1966, pp. 1--23.

Teplyaev, Alexander. "Spectral Zeta Functions of Fractals and the Complex Dynamics of Polynomials." Transactions of the American Mathematical Society, vol. 359, no. 9, 2007, pp. 4339--4358.

Lapidus, Michel L. "Spectral and Fractal Geometry: From the Weyl-Berry Conjecture for the Vibrations of Fractal Drums to the Riemann Zeta-Function." Differential Equations and Mathematical Physics, Mathematical Science Engineering, vol. 186, 1992, pp. 151--182.

Lacunarity and Texture Analysis

Plotnick, Roy E., et al. "Lacunarity Analysis: A General Technique for the Analysis of Spatial Patterns." Physical Review E, vol. 53, no. 5, 1996, pp. 5461--5468.

Allain, Claire, and Michel Cloitre. "Characterizing the Lacunarity of Random and Deterministic Fractal Sets." Physical Review A, vol. 44, no. 6, 1991, pp. 3552--3558.

Gefen, Yuval, Yitzhak Meir, and Amnon Aharony. "Geometric Implementation of Hypercubic Lattices with Noninteger Dimensionality by Use of Low Lacunarity Fractal Lattices." Physical Review Letters, vol. 50, no. 2, 1983, pp. 145--148.