Exploring the Mandelbulb: The Journey to 3D Fractals
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Chapter 1: Introduction to Fractals
In previous discussions, we delved into the history and fundamentals of iterating complex numbers to produce fractals. Starting with Julia Sets, we gradually defined and illustrated the famous Mandelbrot Set, which lies at the core of fractal geometry.
The Mandelbrot Set serves as a comprehensive guide to all Julia Sets. It visually represents how different initial constants, when repeatedly applied in the equation Z² + C, either converge into beautiful, connected Julia Sets or diverge into chaotic Fatou Sets. The diagram below illustrates that selecting initial complex numbers within the Mandelbrot Set generates continuous Julia Sets, while those outside lead to disjointed Fatou Sets.
With our initial questions answered, we found ourselves faced with new queries typical in mathematics: Does a fractal equivalent exist in higher dimensions? Specifically, what is the 3D counterpart of the Mandelbrot Set?
Prologue
It appears that the brilliant Rudy Rucker anticipated this inquiry approximately 33 years ago. As a mathematician, computer scientist, and science fiction writer, Rucker was at the forefront of the STEM field and recognized the significance of the Mandelbrot Set shortly after Benoit Mandelbrot's initial findings. His creative mind pushed him to consider the existence of a 3D structure parallel to the Mandelbrot Set.
Rucker understood the computational limitations of his time, realizing that the billions of calculations needed were virtually unfeasible. Instead of attempting the impossible, he crafted a narrative. In 1987, he penned a short story titled "As Above, So Below," where he imagined discovering the 3D Mandelbrot Set, coining the term "Mandelbulb."
Chapter 2: The Quest for 3D Fractals
The search for a 3D equivalent hinges on the complexities of number systems. The Mandelbrot Set operates in two dimensions due to the two components of complex numbers. The question arises: Can we establish a similar number system for three dimensions?
Section 2.1: Understanding Number Systems
The Mandelbrot iteration (Z² + C) relies on two key operations: addition and squaring of complex numbers. While forming an n-component number system for addition is straightforward (often categorized as a vector space), complications arise when multiplication is introduced.
Creating a vector space with a bilinear product leads to the need for an algebraic structure over a field. To explore the feasibility of a three-dimensional number system, we can start with complex numbers and introduce a third component, j. It becomes essential to ensure that properties of complex and real numbers, such as associativity and commutativity, hold true in this new system.
Section 2.2: The Challenge of Dimensionality
The attempt to construct a functional three-dimensional number system reveals that no natural choice exists. A breakthrough came from Hurwitz's Theorem, which states that for a number system to support certain operations, it must belong to one of four mathematical spaces: real numbers (1D), complex numbers (2D), quaternions (4D), or octonions (8D)—but no 3D systems are available.
In 1982, Alan Norton attempted to find a 3D Mandelbrot equivalent using a quaternion (4D) system. His findings introduced a Quaternion Julia Set by visualizing a 3D "slice" of the 4D space.
The first video, "Mandelbulbs: The Search for a 3D Mandelbrot Fractal," explores the complexities involved in discovering 3D fractals and highlights Norton's contributions.
Section 2.3: The Emergence of the Mandelbulb
After two decades of minimal progress, the quest for the 3D Mandelbrot Set reawakened in 2007 with amateur mathematician Daniel White. His innovative interpretation of the Mandelbrot definition geometrically made it feasible to work in three dimensions. Instead of rotating around a circle, he proposed rotating around spherical coordinates (x, y, z).
White's formula, shared on fractal forums in November 2007, aimed to replicate the escape behavior observed in the 2D Mandelbrot Set by introducing a hyper-complex point (z).
The second video, "The Mandelbulb - 3D Fractal Tour in 8k," visually illustrates the beauty and complexity of the Mandelbulb, showcasing the results of White's approach.
Section 2.4: The Search Continues
Despite the excitement surrounding White's approach, initial results were less than thrilling. Although the outputs resembled the Mandelbrot Set with an additional axis, they lacked the intricate detail expected of a true 3D Mandelbrot.
In 2009, Paul Nylander, along with White, ventured further by experimenting with raising Z to different powers, leading to the discovery of the Mandelbulb, defined by the formula Z⁸ + C.
In conclusion, the Mandelbulb signifies a significant advancement in fractal geometry, possessing many characteristics of the original Mandelbrot Set. It captivates with its infinite complexity, though skepticism remains as the search for the definitive 3D Mandelbrot continues.
While the journey is fraught with challenges, each leap in understanding brings us closer to unraveling the mysteries of fractals in higher dimensions.