Glossary

The guide's working vocabulary, one paragraph each. Chapter links go to the place a term is actually put to work.

Fractal — a shape whose detail never runs out: magnify it and new structure appears at every scale. Coined by Benoît Mandelbrot (1975), from the Latin fractus. (What a fractal is)

Fractal dimension — the box-counting exponent $D$ in $N(\varepsilon) \sim \varepsilon^{-D}$: how fast detail multiplies as the measuring scale shrinks. Whole for classical shapes, fractional for fractals (Koch curve: $\log 4/\log 3 \approx 1.262$); the Mandelbrot set's boundary achieves the plane's maximum, $D = 2$ (Shishikura). (What a fractal is)

Self-similarity — repetition across scales, in three strengths: exact (constructed fractals like the Koch curve), quasi (the Mandelbrot set: copies everywhere, no two identical), statistical (coastlines and clouds: only the roughness statistics repeat). (What a fractal is)

Orbit — the sequence $z_0, z_1, z_2, \dots$ produced by iterating a rule from a starting point. Everything in escape-time rendering is a question about an orbit's fate. (Mandelbrot)

Bounded / escaping — the two fates. A bounded orbit stays within some fixed distance of the origin forever; an escaping orbit grows without limit. For $z \mapsto z^2 + c$, crossing $|z| = 2$ proves escape. (Mandelbrot)

Escape time — the number of iterations an orbit takes to cross the bailout circle; the raw material of every exterior colour. (Escape time)

Bailout — the escape test's threshold. For $z^2 + c$ any radius $\ge 2$ gives the same verdict (other maps have their own escape radius); Mandala uses 256 across the families because the smooth-colouring formula becomes more accurate the further out the orbit is measured. (Escape time)

Smooth (continuous) escape count — the fractional iteration count $\nu = n + 1 - \log_2 \log |z_n|$, which removes the arbitrariness of where the bailout sits and melts colour bands into gradients. (Escape time)

Colour density — the frequency $f$ that scales the smooth count before the palette: how tightly the colour bands pack. The dressing's one main knob; the measurement underneath never changes. (Colour)

Custom gradient / LUT — the user-authored palette: colour stops sampled once into a 256-entry lookup table that every engine indexes with wrap-around, so Rust, GPU and deep-zoom agree pixel-for-pixel. (Colour)

Flow (palette cycling) — animating the palette's phase offset along the contours of constant escape count. Pure dressing: nothing about the fractal is recomputed. (Colour)

Histogram colouring — the classic alternative mapping: colour each pixel by its rank in the whole frame's count distribution. Automatic contrast, but needs every pixel counted before any is painted — which is why a streaming tile renderer doesn't use it. (Colour)

Parameter plane — the picture whose pixels are values of $c$: every pixel is a different rule, iterated from the critical point. The Mandelbrot set lives here. (Julia)

Dynamical plane — the picture whose pixels are starting points $z_0$ under one fixed rule. Julia sets live here. (Julia)

Filled Julia set / Julia set — for a fixed $c$, the set of starting points whose orbits stay bounded; properly, the Julia set is its boundary. Connected exactly when $c$ belongs to the Mandelbrot set. (Julia)

Critical point — a point where the map's derivative vanishes ($z = 0$ for $z^2 + c$). The orbit of the critical point is the diagnostic orbit: its fate decides whether the Julia set is connected or dust. (Julia)

Fixed point / periodic cycle / period — a fixed point satisfies $f(z^*) = z^*$; a period-$k$ cycle returns to its start after $k$ steps. The bulbs of the Mandelbrot set are indexed by the period of the cycle their orbits settle into. (Mandelbrot)

Multiplier — the derivative of the map around a cycle (for a fixed point of $z^2+c$, $\lambda = 2z^*$; for a period-2 cycle, $4z_1 z_2$). The cycle attracts when $|\lambda| < 1$; setting $|\lambda| = 1$ traces the boundary of a cardioid or bulb. (Mandelbrot)

Main cardioid — the heart-shaped body of the Mandelbrot set: parameters whose orbit settles to an attracting fixed point, $c = \lambda/2 - \lambda^2/4$ with $|\lambda| \le 1$. (Mandelbrot)

Bulb — a disk-like component attached to the cardioid where orbits settle into an attracting cycle; the period-2 bulb is the disk $|c+1| < 1/4$. (Mandelbrot)

Misiurewicz point — a parameter whose critical orbit is eventually periodic without being periodic from the start (like $c = i$). They sit on the boundary, in the filaments, and the set is asymptotically self-similar around them. (Mandelbrot)

Minibrot — a miniature copy of the whole Mandelbrot set embedded in its own boundary. Small copies are dense along the boundary — a theorem of Douady–Hubbard renormalization — so magnifying any boundary stretch far enough always reveals one; there are infinitely many, and no two are identical.

Dendrite — a Julia set with branches but no interior, from a seed exactly on the boundary (the classic one is $c = i$). (Julia)

Basin (of attraction) — for a root-finding fractal, the set of starting points that converge to a given root; Newton and Nova colour by basin. (Families)

Analytic (holomorphic) — complex-differentiable, like the polynomial rules of Mandelbrot and Multibrot. The folds in Burning Ship, Tricorn and Celtic break analyticity, which changes both the pictures and the machinery available to render them. (Families)

Derivative recurrence — the extra per-step recurrence $z'_{n+1} = 2 z_n z'_n + 1$ that carries the orbit's sensitivity; the input to distance estimation and slope shading. (Shading)

Distance estimation (DE) — the estimate $d \approx |z_n| \ln |z_n| / |z'_n|$ of a pixel's true distance to the set, honest up to a small constant factor; darkening by $d$ inks the boundary. (Shading)

Double-double — arithmetic on unevaluated pairs of doubles whose sum carries ~31 significant digits; the middle rung of the precision ladder. (Deep zoom)

Double-single — a pair of 32-bit GPU floats used to emulate roughly 14 decimal digits of coordinate precision in WGSL. A startup probe detects drivers that collapse the pair and moves rendering to CPU earlier. (Performance)

Perturbation — the deep-zoom technique: one arbitrary-precision reference orbit at the view centre, and per-pixel deltas $\Delta_{n+1} = 2 Z_n \Delta_n + \Delta_n^2 + \delta$ small enough for plain doubles. (Deep zoom)

Reference orbit — the single high-precision orbit that perturbation measures every pixel against. (Deep zoom)

Rebasing — re-pointing a pixel's delta at the start of the reference orbit when the two drift apart; heals perturbation "glitches" without computing new references. (Deep zoom)

BLA (bivariate linear approximation) — precomputed linear steps that approximate $\Delta' = A\Delta+B\delta$ within a conservative validity radius. Power-of-two blocks compose into a binary table, letting one safe lookup skip long runs of perturbation iterations. Also called bilinear approximation in some fractal literature. (Deep zoom)

Floatexp — an extended-range value $m2^e$ used while a deep pixel's delta is too tiny for the exponent range of an IEEE double. Once the delta grows into normal range, rendering hands back to the fast $f64$ perturbation loop. (Deep zoom)

Tile — the unit of parallel CPU rendering: the frame is cut into small squares dealt to a worker pool, painted centre-out as each returns, and cancelled by frame id when you move on. (Performance)

Supersampling — evaluating several complete sub-pixel orbits per output pixel, colouring each independently, then averaging their RGB values. A 2× setting costs four samples; 3× costs nine. (Performance)

Interior rejection — proving a pixel is in the set without iterating: the cardioid and period-2 disk inequalities answer membership in a handful of multiplications, skipping the most expensive pixels on screen. (Performance)

Iteration budget — the per-frame cap on iterations, scaled automatically (roughly with the square of the zoom depth) so boundary pixels aren't falsely painted black at depth. (Performance)

Feigenbaum point — $c \approx -1.401155$, where the period-doubling cascade of bulbs along the real axis ($1 \to 2 \to 4 \to 8 \to \cdots$) accumulates; beyond it lies the antenna out to the tip at $c = -2$. (Mandelbrot)