# Fractal Self-Affinity in Nature > Self-affine fractals, metabolic scaling theory and the geometry of biological systems: tutorial chapters on fractal foundations, four research manuscripts, pre-rendered Jupyter notebooks, interactive apps, and OKF knowledge bundles of fractal concepts and proofs. Related OKF bundle: https://tyson-swetnam.github.io/okf/index.md Related OKF bundle: https://tyson-swetnam.github.io/intro-gpt/llms.txt Related OKF bundle: https://tyson-swetnam.github.io/energetics-pymol/llms.txt Related OKF bundle: https://tyson-swetnam.github.io/cod-kmap/llms.txt The site is an Open Knowledge Format (OKF v0.2) bundle: every page carries YAML frontmatter with type, description, provenance (generated, sources, verified) and lifecycle (status). Each link below is the page's verbatim Markdown source; the rendered page is the same URL with .md replaced by / (directory pages: drop the trailing index.md). The whole corpus is in https://tyson-swetnam.github.io/fractal-notebooks/llms-full.txt. The entries under "Interactive apps" are one React single-page application (HashRouter): every route shares the crawlable URL https://tyson-swetnam.github.io/fractal-notebooks/react/ and needs JavaScript. Read https://tyson-swetnam.github.io/fractal-notebooks/applications/react-apps.md for what each app does. ## Home - [Fractal Patterns in Nature: Self-Affinity vs Self-Similarity](https://tyson-swetnam.github.io/fractal-notebooks/index.md): Home page of Fractal Notebooks: Part I foundations of fractal geometry, Parts II-V research on self-affine scaling in vascular organisms, rendered Jupyter notebooks, React and Streamlit apps, OKF knowledge bundles and a guide for AI agents. ## Foundations - [Part I: Foundations](https://tyson-swetnam.github.io/fractal-notebooks/foundations/index.md): Landing page for Part I: four tutorial chapters on the history of fractal mathematics, dimension theory and measurement methods, fractals from 1D signals to 4D spacetime, and a glossary with the key equations, plus their companion Jupyter notebooks and a recommended reading order. - [Chapter 1: History of Fractal Mathematics](https://tyson-swetnam.github.io/fractal-notebooks/foundations/history.md): Traces fractal geometry from the mathematical monsters of Cantor, Koch, Sierpinski and Julia through Mandelbrot, self-affinity and the Hurst exponent, IFS, L-systems and DLA, metabolic scaling theory, and complex dimensions, with an appendix of machine-checked proofs. - [Chapter 2: Mathematical Foundations of Fractals](https://tyson-swetnam.github.io/fractal-notebooks/foundations/mathematics.md): Develops the tools for measuring fractals: topological versus fractal dimension, Hausdorff-Besicovitch and box-counting dimensions, differential box-counting for grayscale and 3D data, power laws and log-log regression, the Hurst exponent and fractional Brownian motion, and lacunarity. - [Chapter 3: Fractal Dimensionality (1D through 4D)](https://tyson-swetnam.github.io/fractal-notebooks/foundations/dimensionality.md): Surveys fractals by embedding dimension: 1/f noise, fractional Brownian motion and Cantor sets in 1D; Brownian surfaces, Mandelbrot and Julia sets, Sierpinski and Koch constructions in 2D with terrain and image applications; Menger sponge, DLA and branching networks in 3D; growth dynamics in 4D. - [Chapter 4: Glossary and Equations](https://tyson-swetnam.github.io/fractal-notebooks/foundations/glossary.md): Alphabetical definitions of fractal geometry and scaling terms, from allometry and the four Barnsley fern IFS maps to spectral dimension, followed by a summary table and detailed reference for the ten key equations and the dimensions of the classical fractals. ## Research — Part II: Metabolic Scaling - [Part II: Metabolic Scaling Theory and Biological Fractals](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/index.md): Landing page for Part II, an unpublished manuscript arguing that the branching networks of vascular organisms are self-affine rather than self-similar, with metabolic scaling theory predictions tested by differential box-counting: table of contents, supporting notebooks and authors. - [Metabolic Scaling — Abstract](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/abstract.md): Abstract and key findings of the Part II manuscript: the ecological literature misdescribes vascular branching networks as self-similar, metabolic scaling theory implies they are self-affine, and a mechanistic theory of fractal dimension is proposed from single cells to multicellular life. (draft) - [Metabolic Scaling — Introduction](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/introduction.md): Introduction to the Part II manuscript: allometry versus isometry, Mandelbrot's self-affine fractals and 1/f noises, why self-similar dimension estimates misreport vascular plants, and Table 1 surveying fractal dimensions reported for plants and forests since Rubner 1883. (draft) - [Metabolic Scaling — Methods](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/methods.md): Methods for Part II: metabolic scaling theory (WBE), the power-law allometric form, radius and length branching ratios n^(-1/2) and n^(-1/3), volume and mass dimension predictions (Equations 9-17) giving d = 3/2, and the FracLac differential box-counting procedure. (draft) - [Metabolic Scaling — Results](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/results.md): Results for Part II: FracLac box-counting and differential box-counting mass dimensions for synthetic L-system trees (Table 1), x-ray leaves (Table 2), branches and roots (Table 3) and LiDAR forest canopy height models (Table 4), all close to the MST prediction d = 3/2. (draft) - [Metabolic Scaling — Discussion](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/discussion.md): Discussion for Part II: asymmetric branching in Bentley et al. and Smith et al. as unrecognised self-affinity, lacunarity, Seely and Macklem's entropy-production hypotheses, Kleiber's 3/4 law, DLA growth of non-vascular organisms, and steps toward a unified fractal theory of ecology. (draft) - [Metabolic Scaling — Conclusion](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/conclusion.md): Conclusion of Part II: natural fractals are mostly self-affine, 1/f spectral noise and organismal fractal dimensions follow from anisotropic scaling, entropy maximisation through mass transfer is the proposed mechanism, with implications for ecology and future research directions. (draft) - [Metabolic Scaling — References](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/references.md): Reference list for the Part II manuscript, grouped by metabolic scaling theory, Mandelbrot's fractal geometry, allometry and historical background, fractal analysis methods, ecological applications, and thermodynamics and entropy. (draft) ## Research — Part III: Spectral Geometry - [Part III: Complex Dimensions and Spectral Geometry](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/index.md): Landing page for Part III, a chapter draft on complex dimensions of fractal strings, the Riemann zeta (Zipf) distribution, Mellin-transform oscillations in divide-and-conquer algorithms, Apollonian packing and R-tree performance on fractal data: summary and section guide. - [Spectral Geometry — Abstract](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/abstract.md): Abstract of the Part III chapter: fractal self-affinity and the Riemann zeta distribution in sorting and 2D packing, complex dimensions of fractal strings, Mellin transform analysis of recurrence wobble, the Apollonian gasket and Zagier's theorem, and a proposed Fractal Riemann Hypothesis. (draft) - [Spectral Geometry — Introduction](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/introduction.md): Introduction to Part III: spectral duality and Kac's drum, why uniformity assumptions fail for Zipfian and fractal data, resonance between recursive algorithms and clustered inputs, the Lapidus-van Frankenhuijsen theory of complex dimensions, and six research objectives. (draft) - [Spectral Geometry — Methods](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/methods.md): Methods for Part III: fractal strings and the Cantor string, geometric zeta functions whose poles are complex dimensions, explicit tube formulas and log-periodic oscillations, fractal sprays, and the Riemann zeta (Zipf) distribution as the probabilistic dual of self-affinity. (draft) - [Spectral Geometry — Results](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/results.md): Results for Part III: limits of the Master Theorem, Flajolet's Mellin transform method and the periodic wobble in divide-and-conquer runtimes, the Apollonian gasket with Hausdorff dimension 1.30568, Zagier's theorem tying packing density to the Riemann Hypothesis, and spectral zeta factorisation. (draft) - [Spectral Geometry — Discussion](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/discussion.md): Discussion for Part III: the Faloutsos-Kamel fractal-dimension cost model for R-trees, a cross-domain table of oscillatory phenomena, lacunarity and the music of the primes, GUE statistics and the Montgomery-Odlyzko law, forest canopies as spectral DNA, and the Fractal Riemann Hypothesis. (draft) - [Spectral Geometry — Conclusion](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/conclusion.md): Conclusion of Part III: five findings linking Zipf-distributed inputs, recursive algorithms, complex dimensions and Riemann zeros, implications of a Fractal Riemann Hypothesis for algorithmic stability, and an outlook on zeta-aware algorithms and quantum computing. (draft) - [Spectral Geometry — References](https://tyson-swetnam.github.io/fractal-notebooks/spectral-geometry/references.md): Works cited for the Part III chapter: Lapidus's fractal strings and complex dimensions, Flajolet's Mellin transform asymptotics, Faloutsos-Kamel R-tree analysis, Apollonian gaskets and the Riemann Hypothesis, and Zipf's law, with web sources accessed in December 2025. (draft) ## Research — Part IV: Biological Geometry - [Part IV: Biological Space-Filling and Thermodynamic Geometry](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/index.md): Landing page for Part IV, a chapter draft extending the WBE model with complex dimensions: self-affine vascular branching, Barnsley-fern iterated function systems, diffusion-limited aggregation in lichens and kelp, crown shyness and Apollonian canopy packing, and log-periodic growth oscillations. - [Biological Geometry — Abstract](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/abstract.md): Abstract of the Part IV chapter: evolution solves biological space-filling with self-affine fractals, anisotropic WBE branching ratios, iterated function systems, diffusion-limited aggregation and competitive packing, and complex dimensions linking growth oscillations to Riemann zeta zeros. (draft) - [Biological Geometry — Introduction](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/introduction.md): Introduction to Part IV: the optimisation problem of capturing diffuse resources at minimal tissue cost, self-affine versus self-similar structures under anisotropic forces, pre-fractals with upper and lower cutoffs and dimension crossover, and the organisation of the chapter. (draft) - [Biological Geometry — Methods](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/methods.md): Methods for Part IV: WBE area-preserving and space-filling branching ratios, the four affine maps of the Barnsley fern as genetic compression, a partition-function formalism for leaf venation, diffusion-limited aggregation in lichens, stromatolites and kelp, and spectral analysis methods. (draft) - [Biological Geometry — Results](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/results.md): Results for Part IV: log-periodic wiggles as complex-dimension signatures, lichen DLA morphology, hydrodynamic DLA in kelp forests, crown shyness as weighted Voronoi tessellation with zeta-distributed gaps, Apollonian canopy packing, preferential attachment and root spectral dimensions. (draft) - [Biological Geometry — Discussion](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/discussion.md): Discussion for Part IV: complex dimensions and log-periodic growth oscillations, the Riemann zeta function and biological rhythms, prime-numbered cicada cycles, phyllotaxis and the golden mean as a spectral optimum, implications for biological scaling theory, and open questions. (draft) - [Biological Geometry — Conclusion](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/conclusion.md): Conclusion of Part IV: a summary table of fractal strategies in biology, self-affinity over self-similarity, log-periodic oscillations as signal rather than noise, organisms as zeta-function machines, and future directions from comparative complex-dimension spectra to synthetic biology. (draft) - [Biological Geometry — References](https://tyson-swetnam.github.io/fractal-notebooks/biological-geometry/references.md): 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. (draft) ## Research — Part V: Research Hypotheses - [Part V: Research Framework and Testable Hypotheses](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/index.md): Landing page for Part V, a preregistration framework testing whether old-growth forests at self-organized criticality show distinct fractal dimension, lacunarity, zeta-distributed gap sizes and GUE-like tree spacing in LiDAR canopy height models: summary, section guide and implementation path. - [Research Hypotheses — Abstract](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/abstract.md): Abstract of the Part V preregistration: five primary hypotheses (optimal filling, scale invariance, zeta gap distribution, universal repulsion, biotic decoupling) and four fractal-string spatial hypotheses for old-growth canopies, with the LiDAR, DBC and statistical methods to test them. (draft) - [Research Hypotheses — Introduction](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/introduction.md): Introduction to Part V: canopy height models as stochastic surfaces, the zeta(2) harmonic packing connection, fractal dimension versus lacunarity, the Montgomery-Odlyzko law and GUE repulsion, statements of the five primary and four fractal-string hypotheses, and chapter organisation. (draft) - [Research Hypotheses — Methods](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/methods.md): Methods for Part V: LiDAR CHM requirements and nCHM normalisation, three forest groups, differential box-counting, gliding-box lacunarity, gap-size power-law fitting, nearest-neighbour spacing against Wigner-Dyson and Poisson, geographically weighted regression, and the analysis plan. (draft) - [Research Hypotheses — Expected Results](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/results.md): Expected results for Part V: predicted fractal dimension, lacunarity R-squared, gap-size exponent, nearest-neighbour spacing and topographic correlation ranges for old-growth, late-successional and plantation forests under null and alternative hypotheses, with a table of testable predictions. (draft) - [Research Hypotheses — Discussion](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/discussion.md): Discussion for Part V: the forest as an energy field under maximum entropy production, biomass as exergy, the WBE hydraulic connection, implications for ecological theory, hearing the forest spectrally, topography and multifractality, thermodynamic condition metrics, limitations and future work. (draft) - [Research Hypotheses — Conclusion](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/conclusion.md): Conclusion of Part V: summary of the five primary and four fractal-string hypotheses, a table mapping ecological concepts to mathematical analogues, an analytical pipeline for LiDAR canopy height models, experimental priorities from chronosequences to cross-biome tests, and framework integration. (draft) - [Research Hypotheses — References](https://tyson-swetnam.github.io/fractal-notebooks/hypotheses/references.md): Reference list for the Part V preregistration framework, grouped from foundational fractal geometry and diffusion-limited aggregation through metabolic scaling, LiDAR canopy structure, self-organized criticality, zeta distributions, random matrix theory, fractal strings and lacunarity. (draft) ## Notebooks - [Notebooks](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/index.md): How the Jupyter notebooks of this project are published as pre-rendered pages, which conda environments run them, and the catalogue of the 18 rendered notebooks: Part I chapters, Part II methods and results, fractal demonstrations and Fractal RAG examples. - [Research Notebooks](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/research.md): Catalogue of the 73 research notebooks not rendered on this site: 3DEP canopy height models for 18 US forests, Barro Colorado Island and Sierra Nevada lidar, Part II reviewer companions, metabolic scaling tutorials, GPU and Blender DLA work, with execution state and GitHub and nbviewer links. ## Notebooks — Part I chapter notebooks - [Chapter 1: History of Fractal Mathematics (notebook)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/1_history.md): Companion notebook to Chapter 1: constructs the Cantor set, Koch snowflake, Sierpinski triangle and carpet, Weierstrass function, Julia and Mandelbrot sets with NumPy and Matplotlib, and tabulates the similarity dimensions of the classical fractals. - [Chapter 2: Mathematical Foundations of Fractals (notebook)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/2_mathematics.md): Companion notebook to Chapter 2: implements box-counting and differential box-counting, power-law fitting with confidence intervals, fractional Brownian motion with Hurst estimation by rescaled range, and gliding-box lacunarity on synthetic fractals. - [Chapter 3: Fractal Dimensionality (1D through 4D) (notebook)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/3_dimensionality.md): Companion notebook to Chapter 3: generates fractals in 1D (coloured noise, fBm, Cantor set), 2D (fBm surfaces, Mandelbrot and Julia sets, Koch and Sierpinski), 3D (Menger sponge, DLA) and as time-evolving DLA growth, reporting the measured dimension of each. - [Chapter 4: Zeta Space Tilings (notebook)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/4_zeta_space.md): Companion to Chapter 1 section 1.6 (complex dimensions); its own title numbers it Chapter 4. Tiles a plane with circles, squares, triangles and polygons whose areas follow A_i = A_0 / i^p, plots the Riemann zeta function, compares exponents p, and relates the packings to Apollonian gaskets. ## Notebooks — Part II methods notebooks - [Metabolic Scaling Theory and Allometric Relationships](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/methods/1_mst_allometry.md): Supports the Part II Methods sections on MST and allometry: fits a log-log regression to metabolic rate versus body mass across 18 orders of magnitude, recovering the Kleiber 3/4 exponent over the Rubner 2/3 surface rule, and states the self-affine framing. - [Branching Ratios and the Space-Filling Dimension](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/methods/2_branching_ratios.md): Supports the Part II Methods section on branching ratios: tabulates the WBE radius and length ratios for n = 2, 3, 4, verifies the space-filling Hausdorff dimension D = 3, derives the metabolic exponent 3/4 two ways, and draws a WBE bifurcating tree as an L-system. - [Volume and Mass-Dimension Predictions](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/methods/3_volume_mass_dimension.md): Supports the Part II Methods sections on volume and mass-dimension predictions: SymPy walks Eqs. 11 to 18, reproduces the projected branch-volume mass dimension d = 3/2, and reconciles the D = 3, d = 3/2 and 4/3 values that appear across the theory. - [Differential Box-Counting and the FracLac Protocol](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/methods/4_dbc_fraclac.md): Supports the Part II Methods sections on differential box-counting and FracLac: implements Sarkar-Chaudhuri DBC from scratch, validates it on fractional Brownian surfaces of known dimension 3 - H, and reproduces the FracLac box-size ladder with its 0.1 growth factor. ## Notebooks — Part II results notebooks - [Synthetic Fractal Dimensions](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/results/1_synthetic_dimensions.md): Supports Results Table 1: transcribes the published synthetic fractal dimensions, regenerates the H-fractal, Pythagoras tree and Barnsley fern, and recomputes box-counting and differential box-counting mass dimensions plus lacunarity for comparison with the paper. - [Observed Leaf and Branch / Root Mass Dimensions](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/results/2_observed_leaf_branch_root.md): Supports Results Tables 2 and 3: transcribes the published leaf and branch/root mass dimensions, recomputes the group means and standard deviations, and validates the box-counting estimator on synthetic fBm contour and venation proxies near d = 3/2. - [Forest Canopy Dimensions and Statistical Validation](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/results/3_canopy_statistical_validation.md): Supports Results Table 4 and the Statistical Validation section: transcribes the six canopy height model dimensions, pools the leaf, branch/root and canopy groups, and runs one-sample t-tests and TOST equivalence tests against the d = 3/2 prediction. ## Notebooks — Fractal demonstrations - [Diffusion Limited Aggregation (DLA)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/dla.md): Diffusion-limited aggregation with Numba: bacterial colonies, tidal-flat stromatolites, upward-biased bryophytes, five lichen growth forms, L-system fruticose lichens with Murray-law tapering, 3D DLA, and open DLA with environmental constraints, each with figures. - [Foundations of Fractal Geometry (notebook)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/fractals.md): Overview notebook for the Foundations chapters with minimal examples of the Cantor set, Sierpinski triangle, Menger sponge and fractional Brownian motion, a self-similar dimension reference table, and pointers to the detailed chapter notebooks. - [Self-Affine Fractal Generators](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/fractal_generators.md): Interactive ipywidgets generators for self-affine fractals: recursive branching trees drawn in 2D with Matplotlib under sliders for depth, angle, length and width ratios, then 3D trees built from tapered-cylinder meshes in Plotly. Stored PNG fallbacks are shown; the sliders need a live kernel. - [Riemann Zeta Space Tiling (interactive)](https://tyson-swetnam.github.io/fractal-notebooks/notebooks/foundations/zeta_space.md): Interactive ipywidgets version of the Riemann zeta space tiling: fills a plane with circles, squares and triangles whose areas decay as A_0 / i^p. The stored outputs are widget views only, so this page shows the code together with the usage notes. (draft) ## Apps - [Interactive Applications](https://tyson-swetnam.github.io/fractal-notebooks/applications/index.md): Overview of the interactive tools: fifteen React apps (sixteen routes including the home page) for fractals, zeta tilings and branching architectures, twelve Streamlit simulations, a Blender DLA point-cloud generator and a differential box-counting guide, with launch links and run instructions. - [Installation Guide](https://tyson-swetnam.github.io/fractal-notebooks/applications/installation.md): Set up the React apps with Node.js and npm, the Python stack for the Streamlit apps with pip or the streamlit-plotly conda environment, a Jupyter kernel, optional CUDA acceleration and the Docker image, with troubleshooting tips. - [React Interactive Applications](https://tyson-swetnam.github.io/fractal-notebooks/applications/react-apps.md): Guide to the fifteen React apps (sixteen routes including the home page): Mandelbrot and Julia sets, Brownian motion, Game of Life, noise and wave dynamics, L-system plants, Barnsley ferns, fractal and Pythagoras trees, DLA, a 3D tree with roots and zeta space tilings, with deep links. - [Streamlit Applications](https://tyson-swetnam.github.io/fractal-notebooks/applications/streamlit-apps.md): Catalogue of the twelve Streamlit apps in apps/ (bacterial DLA and swarming, Physarum, yeast colony, tree growth rings, cambium and 3D tree-root models), how to run them, and the matplotlib and PIL scripts that render static fractal images and GIFs. - [Differential Box-Counting Tool](https://tyson-swetnam.github.io/fractal-notebooks/applications/dbc.md): How differential box-counting extends box-counting to grayscale images, with FracLac (ImageJ) steps, a NumPy implementation, expected dimensions for canopy height models, tissues and terrain, best practices, and the MST prediction of a 3/2 mass dimension. - [3D Tree and Root Visualization](https://tyson-swetnam.github.io/fractal-notebooks/applications/tree-roots-3d.md): Interactive 3D model of self-affine branching networks built on the WBE radius and length ratios, with crown, root and crown-shyness parameters, biological dimension tables, Three.js implementation notes, export options and worked parameter examples. - [Blender DLA Point Cloud Generator](https://tyson-swetnam.github.io/fractal-notebooks/applications/blender-dla.md): 3D diffusion-limited aggregation in Blender 4.2 with Geometry Nodes and Cycles: setup script, flow-field and particle presets, GPU rendering, point-cloud export to NumPy, PLY, OBJ and CSV, batch rendering and the companion analysis notebook. ## Knowledge - [Knowledge Bundles](https://tyson-swetnam.github.io/fractal-notebooks/okf/index.md): The two Open Knowledge Format v0.2 bundles published with this site: 32 generated fractal concepts in 12 groups and 12 hand-written proofs with machine-checked step tags, with how to read them, how they are produced, and how the Fractal RAG stack indexes the same ontology. ## Knowledge — Fractal RAG - [Fractal RAG Stack](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/index.md): Overview of the self-hosted retrieval-augmented generation stack: Weaviate 1.35 with a text2vec-transformers sidecar indexes 50K Constellate articles plus this site's notebooks and pages, queried through an MCP server by the fractal-explorer subagent. - [Fractal RAG: Local Setup (Laptop, No GPU)](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/local-setup.md): Run the Fractal RAG stack on one machine without a GPU: Docker Compose Weaviate with a CPU bge-small embedder, install the fractal-rag CLI, bootstrap the schema, ingest the repository docs and notebooks, wire Claude Code, and re-index after a reorganisation. - [Fractal RAG: Hosted VM (K3s + GPU)](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/hosted-vm.md): Production deployment of the Fractal RAG stack on a single GPU VM with K3s: NVIDIA device plugin, Weaviate 1.35 and the arctic-embed sidecar manifests, iRODS corpus pull, full ingest, remote MCP over SSH and an operations playbook. - [Fractal RAG: Using the fractal-explorer Agent](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/using-the-agent.md): End-user guide to the read-only fractal-explorer retrieval subagent: the FractalArtifact and Concept collections, how to invoke it from Claude Code, the eight MCP tools, example transcripts, citation discipline, known limits and how to iterate on answer quality. - [Fractal RAG: Troubleshooting](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/troubleshooting.md): Failure modes of the Fractal RAG stack by layer: Weaviate and GPU sidecar pods, schema bootstrap, ingest paths including FRACTAL_RAG_REPO_ROOT and CORPUS_ROOT, MCP wiring in Claude Code, agent quality, and the re-index recipe after a docs reorganisation. ## Knowledge — Fractal RAG — Notebooks - [01 — Quickstart: Connect & First Query](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/notebooks/01_quickstart.md): Code-only walkthrough that connects to the Fractal RAG Weaviate instance with the CLI settings, checks the artifact and concept collections, runs a first hybrid search and fetches one full artifact by UUID. Needs a running stack, so no outputs are stored. - [02 — Hybrid Search Patterns](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/notebooks/02_hybrid_search.md): Code-only tour of the retrieval knobs in the Fractal RAG stack: the target_vector choice (content, title or summary), the alpha keyword-vector balance, source_type and year filters, and near_object expansion from an anchor hit. - [03 — Concept Ontology](https://tyson-swetnam.github.io/fractal-notebooks/fractal-rag/notebooks/03_concept_ontology.md): Code-only walkthrough of the Concept side-collection: lists concepts by domain, searches definitions semantically, follows artifact-to-concept cross-references, and checks tagging quality against the seed.yaml ontology. ## Knowledge — Concepts - [Fractal Notebooks — Concept Knowledge Bundle](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/index.md): Root index of the OKF concept bundle (32 concepts in 12 groups) - [Biological Fractals](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/biological/index.md): Index of the 1 concept page(s) in the Biological Fractals group: Allometry. - [Allometry](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/biological/allometry.md): Study of how parts of an organism scale with overall body size, often following power-law relationships such as Kleiber's 3/4-power metabolic law. - [Classical Fractals](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/classics/index.md): Index of the 4 concept page(s) in the Classical Fractals group: Cantor Set, Koch Curve, Menger Sponge, Sierpinski Triangle. - [Cantor Set](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/classics/cantor-set.md): Classical fractal constructed by iteratively removing the middle third of a line segment. - [Koch Curve](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/classics/koch-curve.md): Continuous, nowhere-differentiable fractal curve built by recursively replacing each line segment with a four-segment 'bump'. - [Menger Sponge](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/classics/menger-sponge.md): Three-dimensional generalization of the Cantor set and Sierpinski carpet obtained by iterative subdivision and removal of central cubes. - [Sierpinski Triangle](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/classics/sierpinski-triangle.md): Self-similar fractal formed by recursively removing the central triangle from an equilateral triangle. - [Complex Dynamics](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/complex-dynamics/index.md): Index of the 2 concept page(s) in the Complex Dynamics group: Julia Set, Mandelbrot Set. - [Julia Set](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/complex-dynamics/julia-set.md): Set of complex numbers for which a given iterated holomorphic function (typically z → z^2 + c for fixed c) has chaotic behavior. - [Mandelbrot Set](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/complex-dynamics/mandelbrot-set.md): Set of complex numbers c for which the orbit of 0 under z → z^2 + c remains bounded. - [Dimension Theory](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/index.md): Index of the 9 concept page(s) in the Dimension Theory group: Box-Counting Dimension, Complex Dimensions, Differential Box-Counting, Fractal Dimension, Hausdorff Dimension, Lacunarity, Multi-Fractal, Open Set Condition, Topological Dimension. - [Box-Counting Dimension](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/box-counting-dimension.md): Estimated fractal dimension obtained by covering a set with boxes of edge epsilon and measuring how the count N(epsilon) scales as epsilon → 0. - [Complex Dimensions](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/complex-dimensions.md): Generalization of the real Hausdorff/box-counting dimension to complex values, used by Lapidus et al. to describe oscillatory geometric and spectral asymptotics of fractal sets. - [Differential Box-Counting](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/differential-box-counting.md): Extension of box-counting for grayscale images that partitions the intensity axis to estimate the fractal dimension of textured surfaces. - [Fractal Dimension](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/fractal-dimension.md): A real-valued (often non-integer) measure of how the detail of a pattern changes with the scale at which it is measured. - [Hausdorff Dimension](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/hausdorff-dimension.md): Rigorous notion of dimension defined as the infimum of d such that the d-dimensional Hausdorff measure of the set is zero. - [Lacunarity](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/lacunarity.md): Measure of how a fractal fills space, quantifying gap-size distribution. - [Multi-Fractal](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/multi-fractal.md): Generalization of fractal geometry in which different regions of a set exhibit different local scaling exponents, characterized by a spectrum f(alpha). - [Open Set Condition](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/open-set-condition.md): Separation hypothesis on an iterated function system: there exists a nonempty bounded open set U with the union of the maps' images of U contained in U and those images pairwise disjoint. - [Topological Dimension](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/topological-dimension.md): Integer-valued dimension defined via open covers (Lebesgue covering dimension). - [Dynamics](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dynamics/index.md): Index of the 1 concept page(s) in the Dynamics group: Attractor. - [Attractor](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dynamics/attractor.md): Set toward which a dynamical system tends; in chaotic systems the attractor often has fractal structure (a 'strange' attractor). - [Foundations](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/foundations/index.md): Index of the 1 concept page(s) in the Foundations group: Power Law. - [Power Law](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/foundations/power-law.md): Functional relationship y = c x^alpha; ubiquitous in fractal scaling, allometry, and self-organized criticality. - [Generative Grammars](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/generative/index.md): Index of the 1 concept page(s) in the Generative Grammars group: L-System. - [L-System](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/generative/l-system.md): Parallel string-rewriting grammar (Lindenmayer system) used to model branching biological structures such as plants, algae, and trees. - [Geometry](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/geometry/index.md): Index of the 3 concept page(s) in the Geometry group: Fractal, Self-Affinity, Self-Similarity. - [Fractal](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/geometry/fractal.md): A complex geometric shape exhibiting self-similarity (or self-affinity) at multiple scales, characterized by a non-integer fractal dimension and often generated by recursive or iterative processes. - [Self-Affinity](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/geometry/self-affinity.md): Property of a fractal that resembles itself under anisotropic scaling (different scaling factors along different axes). - [Self-Similarity](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/geometry/self-similarity.md): Property of a fractal that appears the same (up to uniform isotropic scaling) when zoomed into any of its parts. - [Iterated Function Systems](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/ifs/index.md): Index of the 2 concept page(s) in the Iterated Function Systems group: Barnsley Fern, Iterated Function System. - [Barnsley Fern](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/ifs/barnsley-fern.md): Iterated function system (IFS) of four affine maps that produces a fern-like attractor and is a classic biological-fractal example. - [Iterated Function System](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/ifs/iterated-function-system.md): Finite collection of contraction mappings on a complete metric space whose attractor is a (typically fractal) compact set, by the Hutchinson theorem. - [Number Theory](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/number-theory/index.md): Index of the 1 concept page(s) in the Number Theory group: Riemann Zeta Function. - [Riemann Zeta Function](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/number-theory/riemann-zeta-function.md): Complex analytic function zeta(s) = sum_{n≥1} 1/n^s, central to number theory; its non-trivial zeros lie (conjecturally) on the critical line Re(s) = 1/2. - [Statistical Physics](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/statistical-physics/index.md): Index of the 1 concept page(s) in the Statistical Physics group: Self-Organized Criticality. - [Self-Organized Criticality](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/statistical-physics/self-organized-criticality.md): Property of dynamical systems that naturally evolve toward critical states where small perturbations produce avalanches of all sizes, often with fractal spatial structure. - [Stochastic Processes](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/index.md): Index of the 6 concept page(s) in the Stochastic Processes group: Brownian Motion, Diffusion-Limited Aggregation, Fractional Brownian Motion, Hurst Exponent, Percolation Theory, Pink Noise. - [Brownian Motion](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/brownian-motion.md): Continuous-time stochastic process with stationary independent Gaussian increments. - [Diffusion-Limited Aggregation](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/diffusion-limited-aggregation.md): Stochastic growth process in which particles undergoing Brownian motion cluster together, producing fractal aggregates with characteristic dendritic branching. - [Fractional Brownian Motion](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/fractional-brownian-motion.md): Generalization of Brownian motion governed by a Hurst exponent H in (0,1) that controls long-range dependence and the roughness of sample paths. - [Hurst Exponent](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/hurst-exponent.md): Parameter H in (0,1) characterizing long-range dependence and the roughness/persistence of a time series; relates to fractal dimension via D = 2 - H for 1D signals. - [Percolation Theory](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/percolation-theory.md): Branch of statistical physics that studies connectivity in random media, near the critical threshold producing fractal clusters. - [Pink Noise](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/stochastic/pink-noise.md): Random signal whose power spectral density is inversely proportional to frequency; a 1D fractal noise widely observed in natural systems. ## Knowledge — Proofs - [Fractal Notebooks — Proof Knowledge Bundle](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/index.md): Root index of the OKF proof bundle (12 proofs) - [Barnsley Fern — Box-Counting Dimension (Numerical)](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/barnsley-fern-dimension.md): The Barnsley fern is the attractor of a four-map affine IFS with overlapping, non-similarity maps; its dimension has no closed form and is estimated numerically at D ≈ 1.83. - [Cantor Set — Dimension, Measure, and Cardinality](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/cantor-set-dimension.md): The middle-thirds Cantor set has Hausdorff dimension log 2 / log 3, Lebesgue measure zero, and the cardinality of the continuum. - [Complex Dimensions of the Cantor String](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/complex-dimensions-cantor.md): The geometric zeta function of the Cantor string has poles at omega_k = log 2 / log 3 + i (2 pi k / log 3); the real part is the Minkowski dimension and the imaginary parts encode log-periodic oscillations. - [Fatou-Julia Connectedness Dichotomy](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/julia-connectedness.md): The Julia set J_c of z^2 + c is connected if and only if the critical orbit is bounded (c lies in the Mandelbrot set); otherwise it is totally disconnected Cantor dust. - [Koch Curve — Dimension, Perimeter, and Area](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/koch-curve-dimension.md): The Koch curve has Hausdorff dimension log 4 / log 3, infinite length, and the Koch snowflake encloses finite area equal to 8/5 of the seed triangle. - [Mandelbrot Boundary — Hausdorff Dimension 2 (Shishikura)](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/mandelbrot-boundary-dimension.md): The boundary of the Mandelbrot set has Hausdorff dimension 2 (Shishikura 1991/1998); this is asserted from the theorem and is not machine-verifiable. - [Mandelbrot Escape-Radius Criterion](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/mandelbrot-escape.md): Any orbit of f_c(z) = z^2 + c that exceeds modulus 2 (with |z| at least |c|) diverges to infinity, making radius 2 an exact bailout. - [Menger Sponge — Similarity Dimension](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/menger-sponge-dimension.md): The Menger sponge is self-similar with N=20 sub-cubes at ratio 1/3, giving Hausdorff dimension log 20 / log 3. - [Spectral Exponent of Self-Affine Gaussian Processes](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/self-affine-spectral.md): For fractional Brownian motion the power spectrum obeys beta = 2H + 1; the trace has local (graph) dimension 2 - H crossing over to global dimension 1. - [Sierpinski Triangle and Carpet — Dimensions](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/sierpinski-dimension.md): The Sierpinski triangle has Hausdorff dimension log 3 / log 2 and the Sierpinski carpet log 8 / log 3, both via the open-set condition. - [WBE Derivation of the 3/4-Power Metabolic Scaling Law](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/wbe-quarter-power.md): Area-preserving branching plus space-filling delivery in a fractal transport network yields Kleiber's law, in which metabolic rate B scales as body mass M to the 3/4 power. - [Weierstrass Function - Continuous Everywhere, Differentiable Nowhere](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/weierstrass-nondifferentiability.md): The Weierstrass function converges uniformly (so is continuous) yet is nowhere differentiable when ab exceeds 1 + 3 pi / 2, a condition later relaxed by Hardy to ab at least 1. ## For AI Agents - [For AI Agents](https://tyson-swetnam.github.io/fractal-notebooks/agents.md): How AI agents and harnesses should consume this site: llms.txt and llms-full.txt, Markdown twins with OKF v0.2 frontmatter, the concept and proof knowledge bundles, notebook source links, and the trust signals to read before relying on a page. ## Interactive apps - [Home (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/): Landing page listing all interactive fractal apps. - [Mandelbrot Set (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/mandelbrot): WebGL escape-time renderer with zoom, iteration and colour controls. - [Julia Sets (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/julia): WebGL Julia set explorer for f(z) = z^2 + c with a movable parameter c. - [Brownian Motion (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/brownian): Random-walk and fractional Brownian motion traces with a Hurst-exponent control. - [Conway's Game of Life (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/conway): Cellular automaton on a WebGL grid with density and colour-scheme options. - [Noise Patterns (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/noise): White, pink and brown noise generators with spectra. - [Wave Dynamics (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/waves): Interactive wave superposition and interference patterns. - [Plant Architecture Lab (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/plant-lab): L-system plant engine with presets and parameter sliders. - [Fern Frond 3D (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/fern-frond-3d): Three-dimensional fern frond built from an L-system. - [Diffusion-Limited Aggregation (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/dla): DLA cluster growth with vector-field bias modes. - [Barnsley Ferns (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/ferns): Iterated-function-system fern with adjustable affine maps. - [Fractal Trees (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/trees): Recursive branching trees with angle and length-ratio sliders. - [Pythagoras Tree (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/pythagoras): Recursive Pythagoras tree of squares. - [3D Tree with Roots (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/tree-roots-3d): Three-dimensional tree and root system visualisation. - [Zeta Space Tiling (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/zeta-space-tiling): Plane tiling with shape areas following the Riemann zeta distribution. - [3D Zeta Visualization (app)](https://tyson-swetnam.github.io/fractal-notebooks/react/#/zeta-3d-visualization): Three-dimensional Riemann zeta sphere packing. ## Meta - [Change Log](https://tyson-swetnam.github.io/fractal-notebooks/log.md): Dated history of notable changes to the Fractal Self-Affinity in Nature site: the 2026-10-24 Zensical rebuild with OKF v0.2 publishing and agent endpoints, the 2026-07-13 knowledge bundles, and the 2025-12-14 seven-part restructure. - [Concept Bundle Update Log](https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/log.md): dated change history of the concepts bundle (newest first). - [Proof Bundle Change Log](https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/log.md): dated change history of the proofs bundle (newest first). ## Optional - [Full corpus (llms-full.txt)](https://tyson-swetnam.github.io/fractal-notebooks/llms-full.txt): every page in this outline, Markdown source with OKF frontmatter, in one file. - [Sitemap](https://tyson-swetnam.github.io/fractal-notebooks/sitemap.xml): crawl surface of the rendered site. - [Source repository](https://github.com/tyson-swetnam/fractal-notebooks): Markdown sources, notebooks, OKF bundles, React and Streamlit apps. - [Notebooks on nbviewer](https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/tree/main/docs/notebooks/): rendered views of every notebook, including those not published as pages. - [Raw notebooks](https://github.com/tyson-swetnam/fractal-notebooks/tree/main/docs/notebooks): the .ipynb files. - [Streamlit app sources](https://github.com/tyson-swetnam/fractal-notebooks/tree/main/apps): the Python apps, runnable with `streamlit run`.