3D Fractals: Mandelbulb, Mandelbox & Beyond

For roughly thirty years after Benoit Mandelbrot introduced the world to the fractal geometry of nature, mathematicians and computer artists shared a nagging ambition: push the Mandelbrot set into three dimensions. The 2D set — generated by the deceptively simple iteration z_n+1 = z_n² + c on the complex plane — yields infinite boundary detail, iconic bulging cardioids, and an inexhaustible supply of miniature self-similar copies. Surely, the reasoning went, a 3D equivalent would be even more extraordinary.

The problem is algebraic. Complex numbers live on a 2D plane, so the Mandelbrot iteration maps neatly onto pixels. Three-dimensional space lacks an equivalent algebra: quaternions work in four dimensions, not three, and the algebraic structures that do exist in three dimensions lack the properties needed for a well-behaved Mandelbrot-like iteration. Dimension three, as one researcher put it, is the gap in the sequence — present in physical reality but absent from the algebraic roster.

The solution, when it came, did not arrive from a tenured professor or a well-funded lab. It came from two amateur mathematicians — Daniel White and Paul Nylander — collaborating on an internet forum called Fractal Forums, working through evenings and weekends over two years. What they found in 2009 is now known as the Mandelbulb: the most celebrated object in 3D fractal geometry, and the starting point for a family of three-dimensional fractal forms that has since exploded in variety and richness.

How Was the Mandelbulb Discovered?

Daniel White began experimenting with 3D Mandelbrot extensions in November 2007, posting his progress to Fractal Forums, the premier online community for fractal artists and enthusiasts. His core insight was geometric rather than algebraic: rather than looking for a 3D number system that mirrors complex multiplication, he asked what the geometric operations implicit in complex squaring would look like if performed in spherical rather than polar coordinates.

In two-dimensional polar coordinates, squaring a complex number doubles its angle and squares its radius. White extended this idea to three dimensions: in spherical coordinates, a point is described by a radius r, a polar angle θ, and an azimuthal angle φ. White defined the “nth power” of a 3D vector v as:

vⁿ  :=  rⁿ  ⟨  sin(nθ)cos(nφ), sin(nθ)sin(nφ), cos(nθ)  ⟩

The iteration v  ↦  vⁿ + c is then run in the same way as the Mandelbrot set: points whose orbit stays bounded are inside the object; points whose orbit escapes to infinity are outside, colored by escape speed. Paul Nylander refined the formula and rendered the first compelling images of the power-8 version in late 2009. A New Scientist article entitled “The Mandelbulb: first ‘true’ 3D image of famous fractal” brought the discovery to a global audience.

The power-8 Mandelbulb — f(p) = p⁸ + c — quickly became the canonical form. Its eight-lobed body, honeycombed with increasingly fine tentacles and filaments, strikes an almost organic balance between global structure and microscopic detail. As White himself noted to New Scientist, the form is still not a perfect 3D fractal: “There are still ‘whipped cream’ sections, where there isn’t detail.” The iteration does not preserve conformality the way complex-number multiplication does in 2D, so some surface regions remain smoother than others. But the overall structure — self-similar, inexhaustible, and visually arresting — captured the imagination of the mathematical world immediately.

Several contributors beyond White and Nylander were instrumental in the early development. Thomas Ludwig was the first to render the power-2 Mandelbulb in early 2008. David Makin contributed formula work alongside Nylander. Later, Jérémie Brunet, Krzysztof Marczak, and Johan Andersson extended the software ecosystem that made high-quality Mandelbulb rendering accessible to artists worldwide.

What Is the Mandelbox, and How Does It Differ from the Mandelbulb?

A year after the Mandelbulb made headlines, a different kind of 3D fractal appeared — one with a radically different construction and aesthetic. In 2010, Tom Lowe discovered the Mandelbox and published it at his own website. Where the Mandelbulb is organic and bulbous, the Mandelbox is angular and architectural: box-like, riddled with recursive corridors and chambers that evoke a ruined cathedral, a coral reef, or an alien city depending on the scale parameter chosen.

The Mandelbox is an escape-time fractal generated by iterating a two-stage geometric transformation:

1. Box fold: Each coordinate is reflected back toward the origin if it exceeds a defined limit (±1 by default). Think of folding a strip of paper at a crease — the portion that overshoots folds back in.
2. Sphere fold: Points inside a minimum radius are pushed outward (inverted through the inner sphere); points between the minimum and maximum radius are inverted through the outer sphere. Points beyond the maximum radius are unaffected.

After both folds, the vector is scaled by a parameter s and the initial point c is added: z_n+1 = s · fold(z_n) + c. The standard Mandelbox uses s = 2. Changing this single scale parameter transforms the object dramatically — from a compact box at s = 2 to a wispy, foam-like structure at s = −1.5, a value notable for containing approximations of many other well-known fractals nested inside it.

One crucial distinction sets the Mandelbox apart from both the Mandelbulb and the 2D Mandelbrot set: it is technically a map of continuous Julia sets. Each location within the object corresponds to a different Julia set with its own unique character, meaning the fractal varies qualitatively — not just in scale — as you explore it. This also means the Mandelbox can be defined in any number of dimensions: a 2D Mandelbox, a 4D Mandelbox, and higher-dimensional variants all follow from the same folding logic, making it perhaps the most generalizable member of the Mandelbrot family.

How Are 3D Fractals Actually Rendered on a Computer?

Rendering a 2D fractal like the Mandelbrot set is conceptually straightforward: color each pixel based on the escape behavior of its corresponding complex number. A 3D fractal cannot be rendered this way — it occupies volumetric space, and traditional polygon-based rendering pipelines have nothing to work with because the surface is defined only implicitly, by a mathematical iteration condition, not by a mesh of triangles.

The technique that makes 3D fractal rendering possible is called distance-estimated ray marching, developed and popularized in the fractal community largely through the influential blog series Distance Estimated 3D Fractals by Mikael Hvidtfeldt Christensen (Syntopia). The algorithm works as follows:

1. For each pixel, fire a ray from the camera into the scene.
2. At the current ray position, evaluate a distance estimator — a formula that approximates the minimum distance to the fractal surface from the current point.
3. Step the ray forward by (approximately) that distance. Because you stepped at most as far as the nearest surface, you cannot overshoot into the object.
4. Repeat until the estimated distance falls below a threshold (you have reached the surface) or exceeds a maximum (the ray missed everything and exits the scene).

The elegance of distance estimation is that it renders the true mathematical surface at any scale, not a polygon approximation. Zoom into a Mandelbulb render by a factor of a trillion and the surface remains crisp, because the computation simply runs more iterations to resolve the finer detail. The trade-off is compute time: a single high-resolution Mandelbulb render with deep ambient occlusion and volumetric lighting can take hours on a CPU or minutes on a modern GPU.

Two free software packages dominate the 3D fractal rendering community:

• Mandelbulb 3D (MB3D) — Created by Jesse and contributors at Fractal Forums, based on White and Nylander’s work. Offers a large formula library, professional lighting presets including screen-space ambient occlusion, depth-of-field, and volumetric fog. Supports formula hybridization (stacking and blending multiple formulas per render). The most widely used 3D fractal creation tool by community output volume.
• Mandelbulber — Open-source, created by Krzysztof Marczak. Available on Windows, Linux, and Mac. Supports OpenCL GPU preview. Covers Mandelbulb, Mandelbox, Menger Sponge, Quaternion fractals, IFS, and more. Strong choice for scripting and batch-render automation.

Why Do 3D Fractals Capture the Mathematical Imagination?

Part of the answer is purely aesthetic. The Mandelbulb occupies a peculiar visual register: it looks simultaneously natural and impossible. Its surface resembles coral, or Romanesco broccoli, or an eroded desert formation — textures that human perception is calibrated to recognize as organic and real. Yet it is generated by a handful of arithmetic operations, contains no material, grows from no evolutionary process, and repeats its structure at every scale without limit. The brain processes it as something familiar while simultaneously failing to place it in any known category.

This perceptual puzzle is related to what physicist Richard Taylor at the University of Oregon has documented for 2D fractals: human visual systems appear attuned to fractal patterns found in nature, and mid-complexity fractals consistently reduce physiological stress markers in experimental subjects. Taylor’s research was conducted on 2D images, but the same attunement may extend to three-dimensional forms — and the Mandelbulb, with its natural-seeming surface complexity, may activate precisely that response.

There is also a deeper mathematical fascination. The Mandelbulb and Mandelbox opened a territory that researchers are still actively mapping. Hybrid fractals — built by alternating iterations between two or more formulas within a single render — produce objects with characters alien to either parent. A BulbBox hybrid (alternating Mandelbulb and Mandelbox iterations) generates sea-floor-like formations neither formula produces alone. Generalized quaternion Mandelbrot sets in four dimensions can be sliced through 3D hyperplanes to reveal cross-sections with their own intrinsic fractal character.

Beyond art, recent academic work has begun treating the Mandelbulb as a serious mathematical object. A 2022 paper by Javier Barrallo and Daniel White, published in the journal Fractals and available as a preprint on arXiv, analyzes the algebraic foundation of the Mandelbulb construction, situating it within the theory of commutative unital magmas and clarifying its relationship to the 2D Mandelbrot set. The object that began on a hobbyist forum is acquiring rigorous mathematical scaffolding.

For the mathematical foundation that underlies all of this — how the Mandelbrot set generates its boundary complexity through escape-time iteration on the complex plane — or how fractal dimension quantifies surface roughness across scales, those articles provide the grounding context on which 3D fractal geometry builds.

What Other Notable 3D Fractals Exist Beyond Mandelbulb and Mandelbox?

The Mandelbulb and Mandelbox are landmarks, but the landscape of 3D fractals is considerably broader. Each of the following objects represents a different mathematical principle applied to three-dimensional space:

• Quaternion Julia Sets: Computed in four dimensions and sliced into 3D hyperplanes, these produce smooth, cloud-like or tentacled forms. Because quaternion algebra is well-defined, these are mathematically closer to the “true 3D Mandelbrot” idea than the Mandelbulb — but they lack the intricate surface lacework of White and Nylander’s construction.
• Menger Sponge: A classical construction that iteratively removes cubic sub-volumes from a cube. Unlike the Mandelbulb, it is exactly self-similar and has a precisely calculated fractal dimension of log(20) ÷ log(3) ≈ 2.727. See our full article on the Menger Sponge for the mathematical details of its paradoxical zero-volume, infinite-surface-area properties.
• IFS 3D Fractals: Michael Barnsley’s iterated function system framework extends to three dimensions to produce fractal ferns, trees, and tetrahedra in volumetric space, all generated from sets of affine transformations.
• Strange Attractors in 3D: The Lorenz attractor — the original “butterfly” of chaos theory, tracing the path of a chaotic dynamical system through phase space — is a fractal object with a Hausdorff dimension of approximately 2.06, slightly above a 2D surface. It lives in 3D space but never fills it, endlessly weaving a fractal thread around two unstable equilibria.
• Fractal Terrains: Midpoint displacement and diamond-square algorithms use fractal Brownian motion noise to generate the convincing mountain landscapes ubiquitous in film and game environments since the 1980s. The terrain is a 3D fractal surface, but one designed for visual plausibility rather than mathematical purity.

The common thread across all of these objects is the fractal principle itself: a simple rule, iterated, generating complexity that cannot be compressed into a shorter description. Whether the rule is a complex-number quadratic, a geometric fold, a system of affine transformations, or a set of differential equations, the outcome is structure at every scale — the signature fractals leave on three-dimensional space, as reliably as they leave it on the plane.