References¶

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Works Cited¶
Brown, James H., et al. "The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms." Santa Fe Institute Working Papers, 1999. https://www.santafe.edu/research/results/working-papers/the-fourth-dimension-of-life-fractal-geometry-and-
Brown, James H., et al. "The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms." PubMed, 1999. https://pubmed.ncbi.nlm.nih.gov/10356399/
Chen, Joe P. "Research Page." Colgate University, 2025. https://fractologist.github.io/research/
Dana, M. R. "Apollonian Packing of Circles within Ellipses." Algorithms, vol. 16, no. 3, 2023, article 129. https://www.mdpi.com/1999-4893/16/3/129
Drake, J. B., and D. B. Weishampel. "Multifractal Analysis of Canopy Height Measures in a Longleaf Pine Savanna." Forest Ecology and Management, 2000. https://sciences.ucf.edu/biology/games/wp-content/uploads/sites/13/2012/09/DrakeJBfem00.pdf
Egalin, E. "Simulating and Modeling Lichen Growth." CNRS, 2004. https://perso.liris.cnrs.fr/egalin/Articles/2004-lichen.pdf
Enquist, Brian J. "Scaling Laws of Vascular Trees: Of Form and Function." ResearchGate, 2005. https://www.researchgate.net/publication/7617105_Scaling_laws_of_vascular_trees_Of_form_and_function
Gwynne, E., and J. Miller. "Random Walk on Random Planar Maps: Spectral Dimension, Resistance and Displacement." Annals of Probability, vol. 49, no. 3, 2021, pp. 1471-1525. https://projecteuclid.org/journals/annals-of-probability/volume-49/issue-3/Random-walk-on-random-planar-maps--Spectral-dimension-resistance/10.1214/20-AOP1471.pdf
Hayes, Brian. "A Tisket, a Tasket, an Apollonian Gasket." American Scientist, 2012. https://www.americanscientist.org/article/a-tisket-a-tasket-an-apollonian-gasket
Kenkel, N. C. "Fractals in the Biological Sciences." University of Manitoba, 1996. https://home.cc.umanitoba.ca/~kenkel/pubs/1996.pdf
Lapidus, Michel L. "Fractal Strings, Complex Dimensions and the Spectral Operator: From the Riemann Hypothesis to Phase Transitions and Universality." University of Hawaii Number Theory Seminar, 2012. https://math.hawaii.edu/numbertheory2012/Lapidus-Lecture.pdf
Lhotka, Jan M., and Robert A. Loehle. "Testing Foundations of Biological Scaling Theory Using Automated Measurements of Vascular Networks." ResearchGate, 2015. https://www.researchgate.net/publication/281484267_Testing_Foundations_of_Biological_Scaling_Theory_Using_Automated_Measurements_of_Vascular_Networks
Loehle, Craig. "Crown Cover Is Correlated with Relative Density, Tree Slenderness, and Tree Height in Lodgepole Pine." ResearchGate, 2012. https://www.researchgate.net/publication/233494840_Crown_Cover_Is_Correlated_with_Relative_Density_Tree_Slenderness_and_Tree_Height_in_Lodgepole_Pine
Loehle, Craig, and Bailian Li. "Fractal Analysis in a Systems Biology Approach to Cancer." PMC, 2011. https://pmc.ncbi.nlm.nih.gov/articles/PMC3148285/
Lupo, S., et al. "Living on the Edge: Variation in the Abundance and Demography of a Kelp Forest Epibiont." Diversity, vol. 11, no. 8, 2019, article 120. https://www.mdpi.com/1424-2818/11/8/120
Mandelbrot, Benoit B. The Fractal Geometry of Nature. W.H. Freeman, 1982.
Ortenzio, Nicholas. "Barnsley Fern." Observable, 2020. https://observablehq.com/@nor/barnsley-fern
Perez-Garcia, C., et al. "Scaling Representation for Critical Phenomena." ResearchGate, 1990. https://www.researchgate.net/publication/231090574_Scaling_representation_for_critical_phenomena
Pommerening, Arne. "Crown Plasticity in Scots Pine (Pinus sylvestris L.) as a Strategy of Adaptation to Competition and Environmental." Forest Biometrics Lab, 2017. https://pommerening.org/wiki/images/b/b1/UriaDiezPommerening2017.pdf
Raja, M., et al. "Hybrid Artificial Root Foraging Optimizer Based Multilevel Threshold for Image Segmentation." PMC, 2016. https://pmc.ncbi.nlm.nih.gov/articles/PMC5048097/
Rastgoo, S., and M. Requardt. "Emergent Space-Time via a Geometric Renormalization Method." Semantic Scholar, 2016. https://www.semanticscholar.org/paper/Emergent-space-time-via-a-geometric-renormalization-Rastgoo-Requardt/d96fd2e3ddd9654bb9cf6c5bbf41505e980b381c
Rodriguez-Hertz, F., et al. "The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform." Symmetry, vol. 13, no. 10, 2021, article 1922. https://www.mdpi.com/2073-8994/13/10/1922
Schmid, S. A., et al. "Latitude and Community Diversity Primarily Explain Invasion Patterns of Widespread Invasive Plants in Small, Subtropical Lakes." Ecology and Evolution, 2025. https://www.gri.msstate.edu/publications/docs/2025/03/172952025_schmid_ECOL-EVOL_invasion_ecology.pdf
Scientific Inquirer. "The Big Picture: Nature's Fondness for Fractals as Seen in the Black Spleenwort Fern." 2024. https://scientificinquirer.com/2024/06/03/the-big-picture-natures-fondness-for-fractals-as-seen-in-the-black-spleenwort-fern/
Smith, M. "Stationary Growth and Unique Invariant Harmonic Measure of Cylindrical Diffusion Limited Aggregation." PubMed, 2012. https://pubmed.ncbi.nlm.nih.gov/23006279/
Sornette, D. "Critical Ruptures." arXiv, 2000. https://arxiv.org/pdf/cond-mat/0003478
Stevens, Laura E. "Modeling Stromatolite Formation with Diffusion-Limited Aggregation." ScholarWorks at University of Montana, 2023. https://scholarworks.umt.edu/etd/11766/
Stoyan, D., and H. Stoyan. "Fractal Geometry and Ecology of Lichens." World Scientific, 1993. https://www.worldscientific.com/doi/10.1142/S0218348X9300037X
Turcotte, D. L. "Algorithmic Complexity and Thermodynamics of Fractal Growth Processes." World Scientific, 1997. https://www.worldscientific.com/doi/10.1142/S0218348X97000565
Uria-Diez, J., and A. Pommerening. "Understanding Crown Shyness from a 3-D Perspective." PMC, 2021. https://pmc.ncbi.nlm.nih.gov/articles/PMC8557382/
Valladares, F., et al. "Mutualistic Strategies Minimize Coextinction in Plant-Disperser Networks." Proceedings of the Royal Society B, 2017. https://royalsocietypublishing.org/doi/pdf/10.1098/rspb.2016.2302
Watkins, Matthew. "Related Curiosities." University of Exeter, 2020. https://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/curiosities.htm
Watkins, Matthew. "The 'Golden Mean' in Number Theory." University of Exeter, 2020. https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/goldenmean.htm
West, Geoffrey B., et al. "The Fractal Dimension of Plants." University of Vermont, 1998. https://pdodds.w3.uvm.edu/teaching/courses/2009-08UVM-300/docs/others/1998/west1998a.pdf
Wikipedia. "Apollonian Gasket." 2024. https://en.wikipedia.org/wiki/Apollonian_gasket
Wikipedia. "Barnsley Fern." 2024. https://en.wikipedia.org/wiki/Barnsley_fern
Zhao, M., et al. "Assemblage Structure and Spatial Diversity Patterns of Kelp Forest-Associated Fishes in Southern Patagonia." PLOS ONE, 2021. https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0257662
Zhao, Y., et al. "A High-Resolution Global Map of Giant Kelp (Macrocystis pyrifera) Forests and Intertidal Green Algae (Ulvophyceae) with Sentinel-2 Imagery." Remote Sensing, vol. 12, no. 4, 2020, article 694. https://www.mdpi.com/2072-4292/12/4/694
Zimmerman, Z., et al. "A Triangular Model of Fractal Growth with Application to Adsorptive Spin-Coating of Polymers." PubMed, 2024. https://pubmed.ncbi.nlm.nih.gov/38394129/
Additional Resources¶
Foundational Texts¶
Mandelbrot, Benoit B. The Fractal Geometry of Nature. W.H. Freeman, 1982.
Barnsley, Michael F. Fractals Everywhere. Academic Press, 1988.
Lapidus, Michel L., and Machiel van Frankenhuijsen. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. Springer, 2013.
Biological Scaling¶
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.
Enquist, Brian J., and Karl J. Niklas. "Global Allocation Rules for Patterns of Biomass Partitioning in Seed Plants." Science, vol. 295, no. 5559, 2002, pp. 1517-1520.
Number Theory and Physics¶
Berry, Michael V. "Spectral Zeta Functions for Aharonov-Bohm Quantum Billiards." Journal of Physics A, vol. 19, 1986, pp. 2281-2296. https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry150.pdf
Connes, Alain. "Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function." Selecta Mathematica, vol. 5, 1999, pp. 29-106.