Skip to content

Resources Bibliography

Jump to bottom
Bradley Brown edited this page Aug 13, 2026 · 1 revision

Resources & Bibliography

Every piece of mathematics, every algorithm, every UI affordance in Fracturing Fog grew out of work done by someone else. This page collects the references that informed the project — books, papers, canonical websites, GitHub repositories, library docs, and useful explainers. Where the application ships a particular algorithm or formula, you should be able to trace it from the relevant doc page back to a citation here.

If you spot a missing reference, file an issue or open a pull request. Citations age — broken links are flagged for fixing in Documentation Plan → Citation maintenance.

Foundational mathematics

Fractal geometry and complex dynamics

  • Benoit B. Mandelbrot. The Fractal Geometry of Nature. W. H. Freeman, 1982. The book that coined the word "fractal" and gave the Mandelbrot set its first wide audience.
  • Adrien Douady, John H. Hubbard. Étude dynamique des polynômes complexes. Publications mathématiques d'Orsay, 1984/1985. Proved the Mandelbrot set is connected; introduced the parameter-ray theory used to label hyperbolic components.
  • Heinz-Otto Peitgen, Peter H. Richter. The Beauty of Fractals: Images of Complex Dynamical Systems. Springer-Verlag, 1986. The earliest accessible visual catalogue.
  • John W. Milnor. Dynamics in One Complex Variable. Princeton University Press, 3rd ed., 2006. The modern standard reference for iterating rational maps on the Riemann sphere.
  • Wolf Jung. Mandel — Software for Complex Dynamics. https://mndynamics.com/indexp.html. Reference tool for orbit-portrait and external-ray verification.

Specific fractal families

  • Pierre Fatou, Gaston Julia. Sur les équations fonctionnelles. Bull. Soc. Math. France, 1917-1919. The origins of complex iteration theory; the Julia sets are named here.
  • Michael F. Barnsley. Fractals Everywhere. Academic Press, 2nd ed., 1993. Defines the chaos game and the iterated function systems used by the IFS renderer.
  • John Hutchinson. Fractals and self similarity. Indiana University Mathematics Journal, 1981. Original IFS attractor theorem.
  • Shigehiro Ushiki. Phoenix. IEEE Transactions on Circuits and Systems, 1988. The two-step memory recurrence used by the Phoenix family.
  • Michael Michelitsch, Otto E. Rössler. A New Feature in the Mandelbrot Set. Computers & Graphics, 1992. Defines the Burning Ship.
  • Daniel White. Triplex algebra and the Mandelbulb. https://www.skytopia.com/project/fractal/2mandelbulb.html, 2007-2009. The Mandelbulb's triplex-power formula.
  • Paul Nylander. Hypercomplex fractals gallery and notes. https://www.bugman123.com/Hypercomplex/. Companion to White's Mandelbulb explorations.
  • Melinda Green. Buddhabrot rendering technique. http://superliminal.com/fractals/bbrot/, 1993.
  • Clifford A. Pickover. Computers, Pattern, Chaos, and Beauty. St. Martin's Press, 1990. Source of the Clifford / De Jong attractor formulae used by the Strange Attractor family.

Deep zoom acceleration

  • Kevin I. Martin. Superfractalthing: Mandelbrot Set Calculation in High Precision. Bulletin of the Mandel-machine project, 2014. Original published perturbation + series approximation algorithm now widely re-implemented (including in this project).
  • Claude Heiland-Allen. MandelbrotPerturbator implementation notes. https://mathr.co.uk/mandelbrot/book-draft/. Practical writeup of perturbation theory + bilinear approximation, glitch detection, and rebase reference selection.
  • Botond Kósa. KallesFraktaler2 — source and engineering blog. https://github.com/edyoung/kalles-fraktaler-2. The de-facto reference open-source perturbation Mandelbrot renderer.
  • Yuhao Zhu. Higher-precision arithmetic via double-double and quad-double. In particular QD library by Y. Hida, X. S. Li, D. H. Bailey (https://www.davidhbailey.com/dhbsoftware/). Foundation for the DD / QD pipeline.

Distance estimation, normals, shading

  • John Hart. Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer, 1996. The original distance-estimation raymarcher.
  • Inigo Quilez. Distance estimation, ambient occlusion, soft shadows for raymarched fractals. https://iquilezles.org/articles/. The canonical modern primer.
  • Robert L. Cook, Kenneth E. Torrance. A reflectance model for computer graphics. SIGGRAPH 1982. Source for the PBR3D theme kind's microfacet shading.

Colour theory and palettes

  • Garry T. Krollman. CIELAB and CIECAM02 conversions — referenced by the from-image palette k-means sampler. Practical formulae from https://www.brucelindbloom.com/index.html?Eqn_RGB_to_XYZ.html.
  • Cynthia A. Brewer. ColorBrewer 2. https://colorbrewer2.org. Reference for accessible sequential / diverging palettes, used for several built-in themes.
  • Peter Karpov. Improved orbit-trap colouring for Mandelbrot zoom videos. https://inversed.ru/Blog_2.htm. Direct inspiration for the orbit-trap theme kind.
  • NASA Ames perceptual papers on chromostereopsis (red/blue depth cue). Source for the Chromostereopsis theme kind.

Audio analysis (audio-reactive slideshow)

  • Sebastian Böck, Markus Schedl. Maximum filter vibrato suppression for onset detection. Proceedings of DAFx-13, 2013. Spectral-flux beat detection underpins the audio-reactive engine.
  • Brian C. J. Moore. An Introduction to the Psychology of Hearing. Brill, 6th ed., 2012. Source for the band-weighted detector EQ (bass / lo-mid / mid / hi-mid / high) bands.

Software, libraries, and tooling

Direct dependencies (referenced from *.csproj)

Library Purpose Project home
Avalonia 12 Cross-platform UI framework that hosts the active shell https://avaloniaui.net
Avalonia.Controls.ColorPicker Colour swatch / colour wheel in the theme editor https://github.com/AvaloniaUI/Avalonia
ReactiveUI.Avalonia MVVM with observable bindings; the project's primary VM idiom https://www.reactiveui.net
Vortice.Windows Native-friendly D3D11/12 + DXGI bindings https://github.com/amerkoleci/Vortice.Windows
Silk.NET Cross-platform OpenGL / Vulkan / Metal bindings (cross-platform render path) https://github.com/dotnet/Silk.NET
SkiaSharp Software / OpenGL Skia renderer https://github.com/mono/SkiaSharp
ILGPU C# → SPIR-V / PTX / OpenCL JIT for the GPU calculator path https://www.ilgpu.net
Roslyn (Microsoft.CodeAnalysis.CSharp) C# script compilation for User Equation + theme C# export https://github.com/dotnet/roslyn
Markdig Markdown → HTML for the static documentation site https://github.com/xoofx/markdig
NAudio WASAPI loopback + microphone capture for the audio-reactive engine https://github.com/naudio/NAudio
FFmpeg Lossless and visually-lossless video encoding presets https://ffmpeg.org
KaTeX (CDN, web only) LaTeX rendering inside the static doc site https://katex.org
Prism.js (CDN, web only) Syntax highlighting inside the static doc site https://prismjs.com

Algorithm references implemented in code

  • k-means in CIELAB — Stuart P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 1982. Underlies PaletteBuilder.Lib's palette extractor and the theme editor's From Image… button.
  • Gaussian elimination with partial pivoting — Numerical Recipes (Press et al., 3rd ed.). Used in the Newton calculator's polynomial-coefficient mode (roadmap).
  • Smoothstep / quintic Hermite interpolation — Ken Perlin. Improving noise. SIGGRAPH 2002. Used for the slideshow cross-fade and the video-zoom easing curve.

Scene Engine — camera splines, motion blur, cinematic moves

  • Catmull-Rom splines — Edwin Catmull, Raphael Rom. A class of local interpolating splines. In Computer Aided Geometric Design, Academic Press, 1974. The C¹-continuous interpolating spline the camera track uses by default — a curve that passes through every keyframe with tangents derived from the neighbouring keys. Used in CameraTrack.Evaluate and SceneGlobalTrack.Evaluate.
  • Cubic Hermite interpolation / smoothstep — see the Perlin entry above. The Bezier interpolation mode and the per-key EaseInOut reparametrisation both reduce to smoothstep ($u^2(3-2u)$, a cubic Hermite with zero endpoint tangents).
  • Accumulation motion blur & shutter angle — Rob Cook, Loren Carpenter, Edwin Catmull. The Reyes image rendering architecture. SIGGRAPH 1987. Distributes samples across the open-shutter interval and averages them; the offline scene renderer's --motion-blur sub-frame averaging is the same box-filter-over-the-shutter idea. Shutter fraction here is the film "shutter angle" expressed as a fraction of the frame interval (0.5 ≈ a 180° shutter).
  • Dolly zoom (the "Vertigo" / Hitchcock effect) — coined in Alfred Hitchcock's Vertigo (1958), achieved by dollying the camera while zooming the opposite way. The orbit camera exposes distance (dolly) directly; a true field-of-view zoom track is future work (see the roadmap).
  • Reinhard tone mapping — Erik Reinhard, Michael Stark, Peter Shirley, James Ferwerda. Photographic tone reproduction for digital images. SIGGRAPH 2002. The Reinhard / ReinhardExtended per-shot tone-map operators.
  • ACES filmic tone mapping — Academy Color Encoding System, AMPAS. The ACES per-shot tone-map operator's filmic curve.

Sites, communities, and other long-form references

  • fractalforums.com — long-running fractal-rendering community. Many of the perturbation and glitch-detection refinements implemented here were first vetted there.
  • mathr.co.uk — Claude Heiland-Allen's writing on Mandelbrot rendering. Frequent first-stop reference.
  • iquilezles.org — Inigo Quilez's articles on procedural graphics, raymarching, signed distance functions, noise.
  • shadertoy.com — countless cross-reference shaders for the 3-D / distance-estimation pipeline.
  • scratchapixel.com — Wikipedia-style explainer site for ray tracing, sampling, transforms.
  • redblobgames.com — Amit Patel's interactive write-ups (hex grids, A*, easing). Indirect influence on UI animation design.

How to cite

When introducing a piece of mathematics or an algorithm in a doc page, link inline like this:

The smoothed escape count follows [(Linas Vepstas 1997)](../Resources-Bibliography.md#linas-vepstas-smooth-iter).

…and ensure the anchor exists here. If it does not yet, add it as a one-line stub under the right section and let the next pass flesh it out. The cost of a stub is zero; the cost of an unverifiable claim is the trust readers put in the doc.

Clone this wiki locally