Notebooks¶

Jupyter notebooks carry the computational side of this project: the figures of the Part I chapters are produced in them, the Part II manuscript's equations and tables are re-derived in them, and the fractal generators that the text describes are implemented in them. Eighteen notebooks are published here as rendered pages. The research notebooks that are not published (lidar canopy analyses, GPU experiments, reviewer companions, work in progress) are catalogued on Research Notebooks with links to GitHub and nbviewer.

How notebooks are published¶

Pre-rendered from stored outputs. Each page is generated from the committed .ipynb by scripts/render_notebooks.py (nbconvert); nothing executes at build time, so the outputs you see are the ones saved in the notebook on the date given in the page banner.
Static figures only. Matplotlib figures are extracted as images. Plotly figures appear as the PNG snapshot saved with them; where none was saved, a note links to nbviewer. ipywidgets controls (sliders, interactive plots) cannot run in a static page and are replaced by an italic Interactive widget note. A cell that raised an exception is shown as a failure admonition with the exception name and message, not a traceback.
Every page links its source. The banner at the top of each notebook page links the .ipynb on GitHub, on nbviewer and in Google Colab, plus the raw file; the same links are in the page's frontmatter (notebook.source, notebook.raw, notebook.nbviewer, notebook.colab) together with cell, figure and error counts. The notebooks themselves are not hosted on this site.
Pages are committed. The rendered Markdown and the extracted images live in the repository next to the notebooks and are drift-checked in CI, so a notebook change that is not re-rendered fails the build.

Environments¶

Environment file	Env / kernel name	Stack	Used by
`environments/fractal-foundations-gpu.yml`	`fractal-foundations-gpu`	Python 3.10, CUDA 12.2 toolkit, PyTorch, CuPy, PyCUDA, Numba, NumPy, SciPy, Matplotlib, Plotly, ipywidgets, OpenCV	The fractal demonstrations (`dla`, `fractal_generators`, `dbc`) and the GPU research notebooks; targets NVIDIA driver 535.x
`environments/3dep-environment.yml`	`3dep`	Python 3.11, PDAL, GDAL, rasterio, rioxarray, geopandas, rio-cogeo, ipyleaflet, py3dep	The lidar and canopy-height-model notebooks (3DEP, Barro Colorado Island, Sierra Nevada)
`docs/notebooks/streamlit-plotly.yaml`	`streamlit-plotly`	Python 3.11, Streamlit, Plotly, Numba, NumPy, SciPy, Matplotlib, ipywidgets	`fractal_generators` and the Streamlit apps

The Part I chapter notebooks and the Part II methods and results notebooks need only the scientific Python stack (NumPy, SciPy, Matplotlib, SymPy, Numba) and run on CPU in any of these environments or in Colab.

mamba env create -f environments/fractal-foundations-gpu.yml
mamba activate fractal-foundations-gpu
python -m ipykernel install --user --name fractal-foundations-gpu
jupyter lab docs/notebooks

To run a notebook in Google Colab, open its page here and use the Colab link in the banner, or replace github.com with colab.research.google.com/github in its GitHub URL.

Part I chapter notebooks¶

Companion notebooks to the Foundations chapters; the figures are the ones used in the chapters.

Notebook	Chapter	What it does	Status
Chapter 1: History of Fractal Mathematics	History	Constructs the classical fractals in historical order: Cantor set, Koch snowflake, Sierpinski triangle and carpet, Weierstrass function, Julia and Mandelbrot sets, with a summary table of their dimensions	6 figures; executed. One cell raises a `ValueError` in the Sierpinski carpet routine (the figure still renders) and the Weierstrass cell was never run
Chapter 2: Mathematical Foundations of Fractals	Mathematics	Topological versus fractal dimension, box-counting and differential box-counting, power laws and scaling relations, the Hurst exponent and fractional Brownian motion, lacunarity	8 figures; executed
Chapter 3: Fractal Dimensionality (1D through 4D)	Dimensionality	Fractals in one to four dimensions: fBm traces and Cantor dust, fractional Brownian surfaces and tilings, the Menger sponge, DLA and branching, time-evolving fractals	9 figures; executed
Chapter 4: Zeta Space Tilings	History §1.6 (complex dimensions)	The Riemann zeta function, single-shape and multi-shape zeta tilings, the effect of the exponent p, and the connection to Apollonian gaskets	5 figures; executed (CPU only)

Fractal demonstrations¶

Stand-alone notebooks that implement the generators discussed across Part I.

Notebook	What it does	Environment	Status
Diffusion Limited Aggregation (DLA)	Random-walk DLA with Numba, from the Witten-Sander model to its biological look-alikes: E. coli colonies in agar, stromatolites on a tidal flat, branching bryophytes, radial lichens, and L-systems for fruticose lichen morphology	`fractal-foundations-gpu` (runs on CPU)	16 figures; executed. Two Plotly figures and two widgets are shown as placeholders
Foundations of Fractal Geometry	Quick tour of the essential constructions: the Cantor set (1D), Sierpinski triangle (2D) and Menger sponge (3D), fractional Brownian motion as the self-affine example, and a dimension reference table	scientific Python stack	4 figures; re-executed 2026-10-24
Self-Affine Fractal Generators	Recursive branching trees: 2D trees drawn with Matplotlib under ipywidgets sliders for depth, branch angle, length and width factors, then 3D trees built from tapered-cylinder meshes and rendered with Plotly; almost entirely code (one markdown cell)	`streamlit-plotly`	7 figures; executed. Plotly figures are shown as their PNG snapshots, the 7 widgets as placeholders
Riemann Zeta Space Tiling (interactive)	Fills a plane with circles, squares, triangles or random polygons whose areas follow A0 / i^p, so that the total converges to A0 ζ(p); ipywidgets controls for the shape set, plane area, exponent p, shape count and orientation. The interactive precursor of the Chapter 4 notebook	scientific Python stack	widget-only outputs, no static figures (marked `draft`); the sliders are shown as placeholders, so run it locally or read the Chapter 4 notebook for the figures

Part II notebooks¶

Companion notebooks to the Metabolic Scaling manuscript. Every cell ran in order; each computation is tagged [COMPUTED] (reproduced in the notebook) or [ASSERTED] (taken from the literature or the manuscript).

Methods, accompanying the Methods section:

Notebook	What it does	Status
MST and Allometric Relationships	The allometric power law (Eq. 9), Kleiber's ¾ law, and the self-similar versus self-affine framing of the WBE model	1 figure; executed
Branching Ratios and the Space-Filling Dimension	Branching ratios (Eq. 10), the space-filling Hausdorff dimension D = ln n / ln(1/γ), the metabolic exponent θ = D/(D+1) = ¾, and a recursive bifurcating tree with WBE ratios	1 figure; executed
Volume and Mass-Dimension Predictions	Volume relations (Eqs. 11 to 13), the projected-area box-counting argument (Eqs. 14 to 18), and the reconciliation of D = 3, d = 3/2 and 4/3, worked symbolically with SymPy	text output only (symbolic); executed
DBC and the FracLac Protocol	Differential box-counting from scratch (Sarkar and Chaudhuri), fBm surfaces of known dimension D = 3 − H, the FracLac protocol used in the paper, and one log-log DBC regression	1 figure; executed

Results, accompanying the Results section:

Notebook	What it does	Status
Synthetic Fractal Dimensions	Regenerates the synthetic fractals of Table 1 and recomputes their box-counting and mass dimensions and lacunarity against the published values	3 figures; executed
Observed Leaf and Branch / Root Mass Dimensions	Summary statistics for the published Tables 2 and 3, a method demonstration on synthetic proxies, and the observed dimensions against the d = 3/2 prediction	3 figures; executed
Forest Canopy Dimensions and Statistical Validation	Group statistics for Table 4, one-sample t-tests and a TOST equivalence test against d = 3/2, and a combined forest plot	1 figure; executed

Fractal RAG examples¶

Code-only pages: the notebooks were never executed because they need a running Weaviate instance and the fractal_rag package (see Local Setup).

Notebook	What it does
01 — Quickstart: Connect and First Query	Connect to Weaviate, sanity-check the collections, browse the concept ontology, run a first hybrid search and pull the full content of a hit
02 — Hybrid Search Patterns	How `target_vector` and `alpha` shift the keyword and vector balance, filtering by source type and by year, and `near_object` queries for more-like-this retrieval
03 — Concept Ontology	List concepts by domain, search definitions semantically, find every artifact that references a concept, and sanity-check the tagging quality

Research notebooks¶

Seventy-three further notebooks live in the repository but are not rendered here: the 3DEP canopy-height-model workflows for 18 US forest sites, the Barro Colorado Island and Sierra Nevada lidar analyses, the reviewer companions to the Part II manuscript, the five metabolic-scaling tutorials, GPU DLA experiments, the Blender point-cloud tools, and three notebooks that are still work in progress. Research Notebooks lists each one with its environment, execution state, and GitHub and nbviewer links.

Adding a notebook to the site¶

Commit the .ipynb with its outputs saved (figures as PNG; keep interactive outputs to a minimum).
Add a [[notebook]] entry to scripts/notebooks.toml (title, description, tags, companion page, environment, date) and a nav entry in zensical.toml.
Run python scripts/render_notebooks.py and commit the generated page together with its _files/ directory.