--- type: Paper Section title: Biological Geometry — References description: >- Works cited and additional resources for the Part IV chapter: WBE allometric scaling, the Barnsley fern, diffusion-limited aggregation in lichens and stromatolites, crown shyness, kelp forest remote sensing, Lapidus's fractal strings, and log-periodic phenomena. resource: https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/references/ tags: [paper, part-iv, reference, biological] paper: title: "Architectures of the Infinite: Self-Affine Fractals, Complex Dimensions, and the Thermodynamic Geometries of Biological Space-Filling" part_title: Biological Space-Filling and Thermodynamic Geometry landing: https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/ stage: chapter-draft sources: - id: santini2023 resource: https://doi.org/10.3390/a16030129 title: Apollonian packing of circles within ellipses (Santini, Mangini and Frezza 2023, Algorithms) - id: gwynne-miller2021 resource: https://doi.org/10.1214/20-AOP1471 title: "Random walk on random planar maps: spectral dimension, resistance and displacement (Gwynne and Miller 2021, Annals of Probability)" - id: arkema-samhouri2019 resource: https://doi.org/10.3390/d11080120 title: "Living on the edge: variation in the abundance and demography of a kelp forest epibiont (Arkema and Samhouri 2019, Diversity)" - id: altes-chirplets2021 resource: https://doi.org/10.3390/sym13101922 title: The Altes family of log-periodic chirplets and the hyperbolic chirplet transform (Symmetry, 2021) - id: de-marchi-cassi1993 resource: https://doi.org/10.1142/S0218348X9300037X title: Fractal geometry and ecology of lichens (De Marchi and Cassi 1993, Fractals) - id: dewey1997 resource: https://doi.org/10.1142/S0218348X97000565 title: Algorithmic complexity and thermodynamics of fractal growth processes (Dewey 1997, Fractals) generated: by: human:tswetnam at: "2025-12-14T00:00:00Z" status: draft --- # References Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. ## 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-](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/](https://pubmed.ncbi.nlm.nih.gov/10356399/) Chen, Joe P. "Research Page." *Colgate University*, 2025. [https://fractologist.github.io/research/](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](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](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](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](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](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](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](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](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](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](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/](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](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](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](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](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/](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](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](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](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/](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/](https://pubmed.ncbi.nlm.nih.gov/23006279/) Sornette, D. "Critical Ruptures." *arXiv*, 2000. [https://arxiv.org/pdf/cond-mat/0003478](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/](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](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](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/](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](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](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](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](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](https://en.wikipedia.org/wiki/Apollonian_gasket) Wikipedia. "Barnsley Fern." 2024. [https://en.wikipedia.org/wiki/Barnsley_fern](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](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](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/](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](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.