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Mandelbulbs

Short answer: The Mandelbrot set is drawn by repeatedly squaring a complex number and adding a constant, then coloring the points whose values never escape to infinity. Fractal hobbyists spent years looking for a true 3D version.

The obstacle is that there's no three-dimensional number system that works like the complex numbers, so there's no canonical 3D Mandelbrot. Rudy Rucker imagined the idea in the 1980s, and Daniel White published a spherical-coordinate formula on Fractal Forums in 2007. In 2009 Paul Nylander raised the power, usually to 8, and the result, the Mandelbulb, showed buds on buds at every zoom level. It's computed by iterating v → vⁿ + c in spherical coordinates and keeping the points whose orbits stay bounded. The power n sets the number of lobes. Mandelbulb interiors appear in Disney's Big Hero 6 (2014) and Annihilation (2018).

VideoWrite-up3 sources
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What should you know about Mandelbulbs?

Quick facts. Each one stands alone and has a stable ID you can cite.

  1. The Mandelbulb, a 3D cousin of the Mandelbrot set, was found by fractal hobbyists Daniel White and Paul Nylander in 2009.

    DD-047-F01 · Write-up · Exhibit C (Power 8)

  2. There's no true 3D Mandelbrot set, because no three-dimensional number system works like the complex numbers.

    DD-047-F02 · Write-up · Exhibit A (No 3D complex numbers)

Video transcript: Mandelbulbs: The Mandelbrot Set in Three Dimensions

DID YOU KNOW? Mandelbulb fractals are three-dimensional extensions of the famous two-dimensional Mandelbrot set, created to explore complex patterns in higher dimensions. Unlike traditional fractals, which exist on a flat plane, mandelbulbs exhibit intricate, self-similar structures that twist and fold in three-dimensional space, providing shapes that resemble organic forms, towers, and spirals. They are generated using mathematical equations applied iteratively, with small changes producing dramatic variations in the overall shape. These fractals are popular in computer graphics and digital art because of their surreal, otherworldly appearance. NOW YOU KNOW!

Burned-in captions, read frame by frame (see transcript.md) · 32.7s Reel ·

Video coming soonMandelbulbs · Instagram Reel
Mandelbulbs: The Mandelbrot Set in Three Dimensions Instagram Reel · · 32.7s

The write-up

The Mandelbrot set in three dimensions. Found in 2009, the Mandelbulb is the closest thing yet to a 3D version of the famous fractal.

  1. Exhibit A

    No 3D complex numbers

    There's no canonical 3D Mandelbrot set, because there's no three-dimensional analogue of the complex numbers. Four-dimensional versions can be built with quaternions or bicomplex numbers.

  2. Exhibit B

    A hobbyist hunt

    Rudy Rucker imagined a 3D Mandelbrot around 1987, and Daniel White published a spherical-coordinate formula on Fractal Forums in November 2007. The power-2 version showed little true fractal detail.

  3. Exhibit C

    Power 8

    Paul Nylander raised the formula to higher powers. At power 8, deep zooms showed bulbs sprouting smaller bulbs in all three dimensions. White and Nylander's 2009 Mandelbulb was the result.

  4. Exhibit D

    How it's made

    Each point is treated as a vector, raised to the nth power by multiplying its spherical angles by n and raising its radius to the nth power, and then the starting point is added back. Points whose orbits stay bounded belong to the set. For n > 3, n sets the number of lobes, and many renders use n = 8.

  5. Exhibit E

    Not perfect

    Even its finder hedged on whether it's 'the real 3D Mandelbrot'. Some regions show smeared 'whipped cream' textures rather than crisp detail.

  6. Exhibit F

    On screen

    The wormhole climax of Big Hero 6 (2014) is a stylized Mandelbulb interior, and the alien in Annihilation (2018) takes the form of a partial Mandelbulb.

Questions people ask about Mandelbulbs

Why is there no 'true' 3D Mandelbrot set?

Four-dimensional versions exist using quaternions.

Source: Write-up · Exhibit A (No 3D complex numbers) · Mandelbulb (Wikipedia); Daniel White, Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal (Skytopia, 2009)

Which power is used in most Mandelbulb renders?

Power 8 was where deep zooms first showed rich detail.

Source: Write-up · Exhibit C (Power 8) · Daniel White, Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal (Skytopia, 2009); Mandelbulb (Wikipedia)

Who published the 2009 Mandelbulb?

They were fractal hobbyists posting on Fractal Forums.

Source: Write-up · Exhibit C (Power 8) · Daniel White, Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal (Skytopia, 2009); Mandelbulb (Wikipedia)

What coordinate system does the formula use?

The angles are multiplied by n and the radius is raised to the nth power.

Source: Write-up · Exhibit D (How it's made) · Mandelbulb (Wikipedia)

Which Disney film used a Mandelbulb for its wormhole?

The film's climax takes place inside a stylized Mandelbulb.

Source: Write-up · Exhibit F (On screen) · Hutchins et al., Big Hero 6: Into the Portal (ACM SIGGRAPH 2015 Talks); Mandelbulb (Wikipedia)

Test Your Knowledge

All quizzes →

5 questions, all from this case file. Answers are revealed instantly.

  1. Why is there no 'true' 3D Mandelbrot set?

    1. Computers are too slow
    2. No 3D number system works like the complex numbers
    3. It's banned
    4. It would be infinite
    Show the answer

    Answer: B. No 3D number system works like the complex numbers Four-dimensional versions exist using quaternions. Source: Write-up · Exhibit A (No 3D complex numbers) · Mandelbulb (Wikipedia); Daniel White, Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal (Skytopia, 2009)

  2. Which power is used in most Mandelbulb renders?

    1. 2
    2. 3
    3. 8
    4. 100
    Show the answer

    Answer: C. 8 Power 8 was where deep zooms first showed rich detail. Source: Write-up · Exhibit C (Power 8) · Daniel White, Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal (Skytopia, 2009); Mandelbulb (Wikipedia)

  3. Who published the 2009 Mandelbulb?

    1. Benoit Mandelbrot
    2. Daniel White and Paul Nylander
    3. Alan Turing
    4. Pixar
    Show the answer

    Answer: B. Daniel White and Paul Nylander They were fractal hobbyists posting on Fractal Forums. Source: Write-up · Exhibit C (Power 8) · Daniel White, Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal (Skytopia, 2009); Mandelbulb (Wikipedia)

  4. What coordinate system does the formula use?

    1. Cartesian only
    2. Spherical
    3. Polar 2D only
    4. None
    Show the answer

    Answer: B. Spherical The angles are multiplied by n and the radius is raised to the nth power. Source: Write-up · Exhibit D (How it's made) · Mandelbulb (Wikipedia)

  5. Which Disney film used a Mandelbulb for its wormhole?

    1. Frozen
    2. Big Hero 6
    3. Moana
    4. Zootopia
    Show the answer

    Answer: B. Big Hero 6 The film's climax takes place inside a stylized Mandelbulb. Source: Write-up · Exhibit F (On screen) · Hutchins et al., Big Hero 6: Into the Portal (ACM SIGGRAPH 2015 Talks); Mandelbulb (Wikipedia)

Cite this case file

Easterling, Marcus. "Mandelbulbs." The Detail Department (DetailEnjoyer), case file DD-047, last updated Dec 1, 2025. https://detailenjoyer.com/case/mandelbulbs/

Permalink: https://detailenjoyer.com/case/mandelbulbs/ · Last updated

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