Fractals

A picture discovered by iteration. Each point in the complex plane is asked the same question—escape, converge, or linger—and coasts, basins and ghostly densities appear where neighbouring answers diverge under magnification. Nothing is drawn stroke by stroke; the boundary is what the rule repeatedly reveals.

A fractal is not simply a complicated pattern. The term is associated with structures whose detail persists across scales and whose geometry may resist description by ordinary smooth curves, surfaces or integer-dimensional objects. Exact self-similarity is one possible form, but many important fractals are only statistically, asymptotically or approximately self-similar.

The sphere is organised by the complex plane. A complex number

Compact form: z=x+iy

has a real component (x) and an imaginary component (y), so it can be represented as one point on a two-dimensional plane. A rule that sends one complex number to another becomes a dynamical system on that plane.

The common quadratic iteration is

Compact form: z_n+1=z_n^2+c.

The Julia and Mandelbrot modes use the same formula but ask different questions.

For a Julia set, the parameter (c) is fixed and the starting point (z_0) varies across the image. The renderer asks which starting states remain bounded and which escape under repeated application of the rule. The Julia set is the boundary separating qualitatively different orbit behaviour.

For the Mandelbrot set, the starting state is fixed at (z_0=0) and each pixel supplies a different parameter (c). The image is therefore a map of rule parameters rather than a map of initial states for one fixed rule. A point belongs to the ideal Mandelbrot set when its orbit remains bounded forever.

This difference is one of the most important conceptual lessons in the sphere:

Julia mode explores the dynamical plane of one rule; Mandelbrot mode explores the parameter plane of a family of rules.

The current renderer cannot iterate forever. It assigns a maximum number of iterations. If an orbit exceeds the escape threshold before that limit, the point is classified as escaped. If it does not, the renderer treats it as interior for the current approximation.

That last phrase matters. A point that has not escaped after 180 or 320 iterations has not thereby been proved bounded forever. Some exterior points escape only after long transients. Raising Max iterations can reveal structure that a lower cap incorrectly merged into the dark interior.

Escape-time colour is also not set membership. The renderer uses a smoothed estimate of when the orbit crosses the bailout radius to reduce visible colour banding. Bright bands and gradients encode rate of escape. The mathematical boundary does not contain those colours.

The parameter called escapeRadius is a bailout radius (R). The code squares it internally and tests (|z|^2>R^2). The current control label Escape radius² and help text imply that the user supplies (R^2), which is incorrect. This should be fixed before the parameter is highlighted educationally.

The same control is currently shown in Newton mode even though the Newton-basin implementation does not use it. That is a second surface-contract issue: a visible control should never suggest causal power it does not possess.

The Burning Ship mode changes the quadratic recurrence by replacing the real and imaginary components with their absolute values before squaring:

Compact form: z_n+1= \left(|Rez_n|+i|Imz_n|\right)^2+c, \qquad z_0=0.

That small-looking modification destroys complex analyticity and produces a sharply different parameter set. The “ship,” “harbour” and “terrain” language describes visual resemblance only. The engine is classifying boundedness under a recurrence, not simulating fire, rock or water.

The Newton mode asks a different question. Newton’s method for finding a root of a function iterates

Compact form: z_n+1=z_n-(f(z_n))/(f'(z_n)).

For (f(z)=z^3-1), the implementation becomes

Compact form: z_n+1 = z_n-(z_n^3-1)/(3z_n^2).

There are three roots. Each pixel begins at a different (z_0), and the renderer colours it by the root it approaches and the speed of convergence. Broad coloured regions are basins of attraction. Their interwoven boundaries reveal extreme sensitivity: starting points separated by a tiny distance can converge to different roots.

A Newton-basin image is fractal because of its basin boundary, but it is not an escape-time Mandelbrot image. It classifies destination under an iterative numerical algorithm.

The Buddhabrot reverses the usual visual emphasis. It samples many candidate parameters (c), follows the orbit of (z_0=0), discards those that fail to escape within the iteration limit, and accumulates the positions visited by escaping orbits. Brightness therefore records sampled visitation frequency.

The result is not a second Mandelbrot set and not a photograph of a hidden object. It is a stochastic orbit-density rendering of the same quadratic iteration. Increasing Paths / frame improves the progressive sample more quickly; exposure and contrast reshape the tonemapping; optional fade slowly forgets old accumulated visits.

AL.Floys uses three accumulation bands associated with different portions of an orbit before combining them into the final image. Its “orbit ghost” colour mode is an implementation-specific visual interpretation, not a canonical mathematical invariant.

The current Buddhabrot sampler draws candidate (c) values from the current effective view and projects orbit points into that same view. When Auto explorer moves the centre or span, old accumulated density remains unless the fade gradually removes it. The evolving image can therefore become a composite of several moving view windows. This is visually rich but differs from a stationary Buddhabrot accumulated over one fixed sampling domain.

Auto explorer deserves broader qualification. In Mandelbrot, Burning Ship and Newton modes, it drifts the view centre and span. In Julia mode it also perturbs the constant (c). It is not merely a camera tour through a fixed specimen: for Julia fields, it slowly changes the mathematical rule itself.

The Zoom (span) parameter is technically a half-height in the complex plane. Smaller values zoom in. Horizontal extent scales with the viewport aspect ratio. A deep zoom therefore changes the mathematical region sampled by every screen pixel.

Resolution and numerical precision impose real limits. Escape-time modes render to an internal raster controlled by renderScale and capped at 512×512 cells before being scaled to the stage. Pixelation can be smoothed visually, but no detail exists below the internal sampling grid. Floating-point arithmetic also limits meaningful depth at extreme zoom because nearby representable coordinates eventually become indistinguishable.

Fractals are often introduced through self-similarity, but the most general lesson here is rule repetition amplifying boundary information. A simple recurrence partitions a continuum of starting points or parameters into outcomes. Repeated magnification reveals that the partition boundary carries structure at scales far below the original view.

Finally, this sphere is not a conventional time-evolving simulation in every mode. Julia, Mandelbrot, Burning Ship and Newton fields are recomputed static classifications whenever parameters or view change. Buddhabrot is progressive because it accumulates random orbit samples over time. The shared interface should not erase that distinction.

The mathematical theory of iterated complex functions predates computer graphics. In 1918, Gaston Julia published his major memoir on the iteration of rational functions. Pierre Fatou developed closely related results in the same period. Their work established the dynamical distinction between stable regions and sensitive boundaries long before those boundaries could be rendered at high resolution.

Newton’s root-finding method reaches back to the seventeenth century and was later generalised and systematised by Joseph Raphson and others. The modern computer image of Newton basins turns a numerical algorithm into a global dynamical portrait. James Curry, Lucy Garnett and Dennis Sullivan’s 1983 computer experiments on Newton iteration helped establish the intricate geometry and dynamical importance of these basins.

Benoit Mandelbrot’s work in the 1970s gave the term fractal its modern scientific and cultural force. His 1980 article on complex quadratic iteration helped make the parameter set now bearing his name a central object in complex dynamics. Computer rendering transformed abstract iteration theory into a directly explorable landscape.

Michael Michelitsch and Otto Rössler described the Burning Ship and its quasi-Julia sets in 1992. Their operation—taking absolute values of the real and imaginary components before squaring—showed how a tiny formal change could generate a dramatically different family of boundaries.

Melinda Green developed the rendering method now known as the Buddhabrot in 1993. Rather than colouring each parameter by its escape time, she accumulated the paths of escaping Mandelbrot orbits. The method made the trajectory space visible and demonstrated that a familiar recurrence can support radically different observational instruments.

The intellectual sequence is therefore:

iteration theory before images → computer exploration of parameter and dynamical planes → alternative recurrences and basin portraits → orbit-density rendering

Fractals belongs in ecospheres.ai not as a gallery of decorative patterns, but as an interactive history of how repeated rules, classification questions and rendering choices jointly create mathematical images.

Aspect	Current implementation
Entities / field	A raster of complex-plane sample points; Buddhabrot additionally maintains three progressive orbit-visit accumulators.
State	View centre and span, mode parameters, iteration limits, raster buffers and—for Buddhabrot—stochastic accumulation history.
Julia update	(z_{n+1}=z_n^2+c), with pixel as (z_0) and fixed juliaRe + i·juliaIm as (c).
Mandelbrot update	The same quadratic recurrence, with (z_0=0) and pixel as (c).
Burning Ship update	Absolute real and imaginary components are applied before the quadratic step.
Newton update	Newton iteration for the three roots of (z^3-1); pixels are classified by nearest converged root and iteration count.
Buddhabrot update	Randomly sample (c), retain escaping orbits, accumulate each visited (z) into orbit-length colour bands, then log-tone-map.
Time semantics	Escape and Newton modes are static recomputations. Buddhabrot is a progressive Monte Carlo accumulation. Auto explorer changes view over time and slightly changes Julia (c).
Resolution	Internal raster follows renderScale, capped at 512×512 cells, then scaled to the canvas.
Escape approximation	Finite maximum iterations and bailout radius; smooth escape colouring for exterior points.
Randomness	Buddhabrot path sampling and its seeded sequence. Escape-time and Newton images are deterministic for fixed parameters and raster.
Rendering interpretation	Palette, interior shade, exposure, contrast and orbit bands encode classification or sampled density; they are not extra mathematical dimensions.
Public simple controls	Field specimen, Auto explorer, Zoom (span), Palette and Max iterations.
Public tools	Pan and a high-gain pan tool currently labelled Force.
Known surface issues	Escape radius is mislabeled as radius squared, is ignored by Newton mode, and Force does not apply a mathematical force.

The most truthful one-line description is: a finite computational atlas of outcomes under complex iteration.

Boundary proliferation — detail appears faster than ordinary geometry predicts

Zoom into the edge of the Mandelbrot or a Julia field. Bays, filaments and islands appear inside regions that looked smooth at the previous scale. The rule is not adding decorative detail after zoom; the finer classification was already implicit in the recurrence.

Parameter plane versus dynamical plane — similar equation, different question

Compare Mandelbrot and Julia. One varies the rule parameter (c); the other varies the starting state under one fixed (c). Their relationship is deep, but the images do not answer the same question.

Connectedness changes — one constant reorganises an entire Julia world

Small changes in Julia (c) can turn a connected boundary into dust-like disconnected components or reshape lobes and filaments. The parameter is global: it changes the recurrence applied to every pixel.

Slow escape — finite iteration hides exterior structure

Increase Max iterations near a detailed boundary. Areas previously treated as interior may resolve into narrow exterior channels. The image demonstrates the epistemic difference between “not yet escaped” and “bounded forever.”

Basin interleaving — nearby starts reach different roots

In Newton mode, inspect a boundary between root colours. Magnification reveals repeated interpenetration. Numerical convergence is globally organised into basins whose boundaries are far more intricate than the roots themselves.

Orbit ghosts — trajectories create a different portrait of the same rule

In Buddhabrot mode, a faint image accumulates from escaping paths. Bright regions are not set membership; they are places sampled orbits frequently visited before departure.

Nonanalytic transformation — a small rule change creates a different world

Compare Mandelbrot with Burning Ship. Absolute values before squaring impose new symmetries and sharp “ship” structures, showing that generative form is sensitive to the exact algebraic operation.

Rendering versus mathematics — colour can move while membership does not

Change Palette or interior shade. The image can become dramatically different without any orbit changing. Conversely, changing Max iterations can change classification even if the palette remains fixed.

First 30 seconds

1. Load Mandelbrot / Coast survey.
2. Turn Auto explorer off so the view remains fixed during the experiment.
3. Lower Zoom (span) in several stages rather than dragging continuously. After each change, inspect the boundary for newly resolved channels and islands.
4. Raise Max iterations while keeping the same centre and span. Notice whether dark areas near the coast acquire exterior structure.
5. Switch to Julia / Seahorse drift, turn Auto explorer off again, and compare a dynamical-plane boundary with the Mandelbrot parameter-plane coast.
6. Use Pan to place a complex boundary near the centre before zooming further.

Three experiments

Experiment	Question	Do	Watch for	Why it matters
Not escaped is not proved bounded	How does a finite computation approximate an infinite definition?	Mandelbrot / Coast survey → hold view fixed → compare low and high Max iterations.	Exterior channels emerging from regions previously rendered as interior.	Makes computational uncertainty and iteration caps visible.
One recurrence, two planes	What is the difference between changing a start and changing a rule?	Compare a Mandelbrot view with Julia / Seahorse drift; in Advanced, alter Julia (c) slightly.	Global reorganisation of the Julia boundary.	Distinguishes parameter space from state space.
Set image versus orbit-density image	Can the same quadratic rule be observed in fundamentally different ways?	Load Buddhabrot / Ghost nebula accumulation → turn Auto explorer off → let it accumulate → compare with Mandelbrot.	Progressive ghost structure and sampling noise rather than escape-time bands.	Shows that rendering question, not only equation, determines the image.

A fourth experiment is to load Newton / Basin crystal, zoom into a three-colour boundary and increase Max iterations. Watch how basin classification and convergence speed interact.

Parameter	What it really controls	Increase it	Decrease it	What to watch
Field specimen	Selects the mathematical construction and classification question.	Not scalar.	Not scalar.	Escape, convergence and orbit-density modes must be interpreted differently.
Zoom (span)	Half-height of the sampled complex-plane window.	Larger span zooms out.	Smaller span zooms in.	New boundary detail and eventual resolution/precision limits.
Pan Re / Pan Im (advanced or pointer)	Centre of the complex-plane view.	Moves the sampled region.	Moves oppositely.	A new image does not necessarily mean a new rule.
Max iterations	Computational effort allowed before an orbit is provisionally classified.	Resolves slower escape or slower convergence at greater cost.	Faster but coarser classification.	“Interior” regions changing near boundaries.
Julia c (Re / Im) (advanced)	The fixed global parameter of the Julia recurrence.	Direction-specific.	Direction-specific.	Connectedness and morphology changes across the entire field.
Escape radius (advanced; current UI bug)	Bailout radius (R), squared internally for comparison.	Later escape detection beyond a larger threshold.	Earlier threshold crossing.	Label currently says radius² and should be corrected; Newton ignores it.
Render resolution (advanced)	Internal raster scale before enlargement.	Sharper sampling and higher cost.	Softer, faster image.	Numerical detail versus display smoothing.
Paths / frame (Buddhabrot)	Number of random candidate orbits sampled per frame.	Faster convergence of the density estimate.	Slower, noisier emergence.	Sampling quality, not stronger dynamics.
Buddha max iter (Buddhabrot)	Longest orbit considered before it is treated as non-escaping for that sample.	Includes longer escaping paths at higher cost.	Favours short escape histories.	Fine orbit-density structure.
Ghost fade (Buddhabrot)	Per-second decay of accumulated visit counts.	Shorter image memory and more responsiveness to moving view.	More permanent accumulation.	This is renderer memory, not orbit dynamics.
Auto explorer	Time-dependent view drift; additionally changes Julia (c) slightly.	On/off, not magnitude at simple tier.	Off freezes the controlled specimen.	In Julia mode it changes the rule as well as the view.
Palette / interior shade (visual)	Colour mapping of escape or basin classifications.	Style-specific.	Style-specific.	Appearance can change without mathematical state changing.
Exposure / contrast (Buddhabrot visual)	Log-tonemap response of accumulated orbit visits.	Brighter or more contrastive density.	Darker or flatter density.	Not a change in orbit frequency itself.

The best simple experimental pair is Zoom (span) plus Max iterations. One asks for finer spatial sampling; the other asks for longer temporal iteration.

Julia / Seahorse drift

The opening Julia specimen. Auto explorer makes it visually alive, but controlled study should turn Auto explorer off because it also changes the Julia constant.

Mandelbrot / Coast survey

The best pedagogical baseline. It begins near a richly structured boundary rather than the familiar full-set poster, directing attention to scale and finite iteration.

Burning Ship / Iron harbor descent

A close view of the canonical absolute-value recurrence. Use it to show how one algebraic modification changes the parameter plane.

Newton / Basin crystal

The clearest demonstration that fractal boundaries can separate convergence outcomes rather than bounded and escaping orbits. Root colours are classifications, not substances.

Buddhabrot / Ghost nebula accumulation

The strongest demonstration of observation as method. The image emerges progressively from escaping orbit visitation rather than from per-pixel escape time.

Recommended learning order: Mandelbrot / Coast survey → Julia / Seahorse drift → Newton / Basin crystal → Burning Ship / Iron harbor descent → Buddhabrot / Ghost nebula accumulation.

• Pan drags the complex-plane view. It changes the sample window, not the recurrence.
• Force performs the same kind of pan with a strength-dependent multiplier. It does not push, bend or deform the fractal mathematically.
• There is no pointwise brush that changes membership. Every pixel is recomputed from the selected rule and view.
• Zoom is presently controlled by the span slider rather than a dedicated pointer gesture.

The Force tool should be renamed Fast Pan, Drift or removed from this sphere. A scientific interface should not imply a force acting on an immutable mathematical classification.

A rendered fractal is not an infinite object on the screen. The implementation uses a finite raster, finite iteration caps and floating-point coordinates.

A dark point in an escape-time image is not necessarily proved to belong to the ideal set. It may simply not have escaped before the current limit.

Smooth colours are not part of the Mandelbrot, Julia or Burning Ship set. They encode escape time. Newton colours encode destination root and convergence rate. Buddhabrot brightness encodes sampled orbit visitation.

Not every intricate region is exactly self-similar. The sphere should use language such as recurring structure across scales, fractal boundary or self-similar motifs, not claim that every zoom reproduces the whole image exactly.

Mandelbrot and Julia images do not simulate coastlines, organisms, galaxies, weather or geology. Those metaphors describe visual resemblance.

Buddhabrot is not a separate physical nebula and not a hidden photograph inside the Mandelbrot set. It is a Monte Carlo rendering technique applied to escaping quadratic orbits.

Auto explorer is not always a passive camera. In Julia mode it changes (c), and in moving Buddhabrot mode it can mix accumulation from changing sample windows.

The defensible conclusion is:

Repeated nonlinear rules can divide a continuous plane into outcomes whose boundaries retain structure across extraordinary ranges of scale.

Fractals contributes iteration as a generator of inexhaustible boundary. Chaos follows individual states through time; Fractals applies related nonlinear logic across entire fields of possible starts or parameters. It teaches that a tiny rule can contain more spatial structure than a finite image can resolve, and that every visualisation is also a choice about what outcome to classify.

Foundational and primary sources

1. Gaston Julia, “Mémoire sur l’itération des fonctions rationnelles,” Journal de Mathématiques Pures et Appliquées, Series 8, Vol. 1, 47–245 (1918). NUMDAM record and full text.
2. Pierre Fatou, “Sur les équations fonctionnelles,” Bulletin de la Société Mathématique de France 47–48 (1919–1920). Foundational companion work on rational iteration.
3. Benoit B. Mandelbrot, “Fractal Aspects of the Iteration of (z\rightarrow\lambda z(1-z)) for Complex (\lambda) and (z),” Annals of the New York Academy of Sciences 357, 249–259 (1980). DOI: 10.1111/j.1749-6632.1980.tb29690.x.
4. James H. Curry, Lucy Garnett and Dennis Sullivan, “On the Iteration of a Rational Function: Computer Experiments with Newton’s Method,” Communications in Mathematical Physics 91, 267–277 (1983). DOI: 10.1007/BF01211162.
5. Michael Michelitsch and Otto E. Rössler, “The ‘Burning Ship’ and Its Quasi-Julia Sets,” Computers & Graphics 16(4), 435–438 (1992). DOI: 10.1016/0097-8493(92)90032-Q.
6. Melinda Green, Buddhabrot Fractal Rendering Method. Author account of the technique developed in 1993: superliminal.com/fractals/bbrot.

Modern scientific context

7. John Milnor, Dynamics in One Complex Variable, 3rd ed. (Princeton University Press, 2006).
8. Heinz-Otto Peitgen and Peter H. Richter, The Beauty of Fractals (Springer, 1986). Important historical bridge between complex dynamics, computation and visual exploration.