AEON: Universal Dynamics Reactor

§ I

The Instrument

AEON is a program for looking at mathematical forms: fractals, tilings, polytopes and strange attractors. The canvas fills most of the screen, and the controls stay at the edges until you need them.

The one-minute tour. It plays from this site and from the download, with no other service involved.

§ II

Forms shipped
in this box.

Six samples from the catalog. The studio has 241 named forms in total (58 fractal types and 183 sacred-geometry patterns), among them the Mandelbrot set at degree-3 perturbation deep zoom, all 13 Catalan solids, both aperiodic monotiles found in 2023, the Klein quartic and Cayley graphs of the Mathieu groups.

The Seed of Life: seven equal circles in hexagonal symmetry

Seed of LifeSeven circles, hex symmetry

The Sri Yantra: nine interlocking triangles around a central point

Sri YantraNine triangles, central bindu

Maurer Rosen=6 d=71

A Penrose tiling of kites and darts with fivefold symmetry

Penrose TilingKites and darts, fivefold

Mandelbrotz² + c, |z| ≤ 2

A tesseract, the four-dimensional cube, projected into three dimensions

Tesseract4-cube, 3D-projected

§ III

Lineage.

Everything in the studio comes from someone's work. Here are six of the people it comes from.

1611

Johannes Kepler

Discovers the rhombic dodecahedron and sets out the harmonic geometry of the Platonic solids. The Mysterium Cosmographicum maps the inner planets onto the dodecahedron-icosahedron pair.

Harmonices Mundi · 1619

1879

Felix Klein

Defines the Klein quartic and its automorphism group PSL₂(7), achieving the Hurwitz upper bound of 84(g − 1) = 168 symmetries on a genus-3 Riemann surface.

Über die Transformation siebenter Ordnung · Math. Ann. 14

1931

Heinz Hopf

Constructs the fibration S¹ → S³ → S². Each point of the base S² lifts to a great circle of S³; in stereographic projection, the great circles become linked rings, the Hopf invariant.

Über die Abbildungen der dreidimensionalen Sphäre · Math. Ann. 104

1974

Roger Penrose

Constructs the kite-and-dart tiling, which covers the plane with fivefold symmetry and never repeats. Eight years later, Dan Shechtman found the same fivefold order in a real alloy.

The Role of Aesthetics in Pure and Applied Mathematical Research

1980

Benoît Mandelbrot

Runs z² + c on an IBM computer and prints some of the first pictures of the set that now carries his name. He had coined the word fractal five years earlier.

Fractal Aspects of the Iteration of z → λz(1 − z) for Complex λ and z

2023

Smith, Myers, Kaplan, Goodman-Strauss

Find the hat, a single tile that covers the plane only without repeating. Two months later they find the spectre, which does it without using mirror images. A question open for about fifty years was settled in one spring.

An Aperiodic Monotile · arXiv:2303.10798

§ IV

The mathematics.

Eleven of the methods AEON computes, grouped by field. Each card gives the equation the code actually iterates and the paper it comes from.

Escape-time iterationMandelbrot family · 2D escape time

z_n+1 = z_n² + c ; escape at |z| > 2 Each pixel is a point c ∈ ℂ; the GPU kernel iterates until escape or until maxIter. Burning Ship folds |Im(z)| on the imaginary axis; Tricorn conjugates; M3 / M4 raise the exponent to 3 and 4; Phoenix adds a second-order memory term p · z_n−1; Celtic folds |Re(z² )|; Lambda iterates z(1−z); Buffalo and Newton complete the family.

Mandelbrot 1980
Ann. NY Acad. Sci. 357

Deep-zoom perturbation10⁶⁰-zoom interior renders

Δ_n+1 = 2·Z_n·Δ_n + Δ_n² + δ A reference orbit is iterated in 256-bit fixed-point on the CPU; every other pixel is rendered as a perturbation Δ against it. Past 10²⁵ the kernel switches to a degree-2 Taylor series approximation; past 10⁶⁰, degree-3. Glitch detection rebases the reference when cancellation collapses |Z + Δ| toward zero (Pauldelbrot's criterion). Burning Ship, Celtic and Phoenix carry their own δ-recurrences past 10¹³ (fold-exact |·| arithmetic; Phoenix's memory term tracked exactly), and every integer power 2–8 of the z-family gets the generalized binomial recurrence.

Pauldelbrot 2014
fractalforums.com

Newton iterationroots of unity, 5 fractals

z_n+1 = z_n − f(z_n) / f'(z_n) For f(z) = zⁿ − 1 the iteration converges to one of n roots of unity; each pixel colored by which root it lands on, lightness encoded as iteration count. The Cayley variant uses a non-rational f with three real fixed points and a basin-of-attraction structure first studied by Hubbard & Schleicher.

Hubbard–Schleicher 1986
Compl. Var. Theory Appl. 8

Descartes circle theoremApollonian gasket

k₄ = k₁+k₂+k₃ ± 2√(k₁k₂ + k₂k₃ + k₃k₁) Curvature k = 1/r; outer container has k < 0. The complex-valued Descartes formula recovers the new circle's center z₄; the recursion is depth-capped and produces a residual set of Hausdorff dimension 1.30568673…, the Apollonian dimension computed by McMullen via thermodynamic formalism.

Soddy 1936
McMullen 1998

Schläfli regular polytopes3D + 4D, 46 patterns

Schläfli symbols: {3,3} … {3,3,3,3}
600-cell  {3,3,5} :  120 V, 720 E, 1200 F, 600 C
120-cell  {5,3,3} :  600 V, 1200 E, 720 F, 120 C

The five Platonic solids extend through the thirteen Archimedean and thirteen Catalan duals to the six convex regular 4-polytopes (5-cell, 8-cell, 16-cell, 24-cell, 120-cell, 600-cell), then to nine of the ten Schläfli–Hess regular star polytopes. Vertex sets from Coxeter Table I; stereographic S³ → ℝ³, orthographic ℝ³ → ℝ².

Schläfli 1852/1901
Coxeter 1947

Golden-angle phyllotaxisspiral lattices

θ_i = i · π(3 − √5)   ≈   137.50776° · i
r_i = R · √(i / N)

The divergence angle is the irrational angle most resistant to rational approximation, the inverse of the golden ratio in radians per turn. Vogel's parametrisation distributes N seeds on a disc with the densest packing achievable without translational symmetry; Fibonacci spiral arms emerge spontaneously from the parastichy bands.

Vogel 1979
Math. Biosciences 44

Robinson triangle deflationPenrose tilings · Hat · Spectre

acute  → 1 acute  + 1 obtuse
obtuse → 2 obtuse + 1 acute
ratio  : φ = (1+√5)/2

The Penrose P2 (kite-and-dart) and P3 (rhombi) tilings emerge from φ-self-similar substitution on the two Robinson half-triangles, generating five-fold-symmetric aperiodic tilings of the plane. The Hat (March 2023) and Spectre (May 2023) closed the einstein problem: a single tile that tiles the plane only aperiodically.

Penrose 1974 · Smith,
Myers, Kaplan,
Goodman-Strauss 2023

Möbius dynamicsIndra's Pearls · Kleinian groups

z' = (μz + 1) / (z + μ̄)
μ ∈ ℂ  on the Maskit slice

The two-generator Kleinian group acts on the Riemann sphere via Möbius transformations; the fixed-point pair orbit closes onto the limit set, a self-similar Cantor-set fractal. Maskit's slice μ parameterises the discrete-faithful representations; pearls render as iterated images of seed circles.

Mumford, Series,
Wright 2002

Reaction-diffusionGray-Scott · Gierer-Meinhardt

u_t = D_u∇²u − uv² + F(1 − u)
v_t = D_v∇²v + uv² − (F + k)v

Two-species reaction-diffusion on a torus. The control plane (F, k) partitions into Pearson's empirical regions: spots, stripes, mitosis, solitons, chaos. Iterated through finite-difference Laplacian on a workqueue grid; the fixed initial seed evolves toward Turing-pattern attractors.

Pearson 1993
Science 261

Lenia continuous CAsolitons in continuum

A_t+Δt = clip( A_t + Δt · G(K * A_t) ) Conway's Game of Life with three discrete axes (space, time, state) replaced by their continuous limits. Smooth growth function G applied to a smooth-kernel convolution K * A. The expanded state space hosts hundreds of named solitons (orbiums, gliders, spinners) that emerge spontaneously from rule space.

Bert Chan 2018
arXiv:1812.05433

Strange-attractor flowsLorenz · Hénon · Rössler · Clifford · de Jong · Dadras

ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz

Strange attractors, continuous 3D flows integrated via Runge–Kutta 4 (the Lorenz butterfly's twin lobes, the Rössler band's single fold) and discrete iterated maps (the Hénon quadratic crescent). The orbit never closes; long integration draws each attractor's characteristic geometry. Eighteen ship, each one's parameter space exposed as sliders.

Lorenz 1963
J. Atmos. Sci. 20

§ V

Specimen card.

The technical details: file size, renderers, formats, accessibility and what we promise about it.

AEON · v7.2.0

Renderers: WebGL2 (escape-time + 3D raymarched) · Canvas 2D (geometry, IFS, attractors) · two web workers (geometry off-thread + CPU fractal off-thread)
Deep zoom: Degree-3 Series Approximation past 10⁶⁰ · multi-pass GPU glitch correction · Pauldelbrot rebase · whole-catalog 64-bit sharp zoom to 10¹³, per-type perturbation past it (Burning Ship, Celtic, Phoenix, Multibrots, integer powers 2–8)
Catalog: 58 fractals · 183 sacred-geometry patterns · 51 palettes · 102 fractal-flame variations · 4D polytopes · Mathieu groups · Catalan solids
Interop: Apophysis 7X .flame round-trip (with motion keyframes, HSV palettes, finalxforms) · Adobe Cube .cube LUT import · UGR / MAP / Adobe .ase / GIMP .gpl palette interchange · PNG metadata · frame-exact video export (MP4/WebM)
Accessibility: WCAG 2.1 AA (axe-verified) · keyboard-rebindable · reduced-motion respected · 44×44 touch targets · high-contrast mode · screen-reader friendly
Build envelope: Zero npm dependencies at runtime · zero external HTTP origins · single-file deliverable · CSP script-src SHA-256 hash hardening · fast, standard and full test tiers plus a lint suite, run before every release (unit, consistency, math, mutation, fuzz, parity, smoke, UI, integration, render audit, worker parity, visual, a11y, WebGPU, deep zoom, user journeys)
Contract: Open the HTML file and the app runs, with nothing to install and no network needed. It will keep working for as long as browsers can open a web page.