Mandelbrot sets rendered in a mutually orthogonal configuration of 3 complex planes

Comprehension of Requisite Variety via Rotation of the Complex Plane (Part #2)

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The primary concern of this argument is the identification of aids to comprehension of complexity and coherence, as previously argued (In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007).

In mathematics, the complex plane is a geometric representation of the complex numbers established by the horizontal real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis. Mathematically, the Mandelbrot set (M) is just a set of complex numbers. A given complex number c either belongs to M or it does not. This is admirably explained by Ben Sparks (What's so special about the Mandelbrot Set? 2019), notably with regard to the following using different values of c. The distinctive configurations can be understood as indicative of articulations of distinctive strategies in "strategic space" as mentioned above.

Selected iteration orbits within a Mandelbrot set rendering (selection lmited as indicated by movement of positions of yellow circle)

Sequence produced from the (Mandelbrot Iteration Orbits, GeoGebra) developed by Ben Sparks.

As shown below left, a picture of the Mandelbrot set can be made by coloring black all the points c that belong to M, and colouring white all other points that do not belong to that set.

The fractal boundary of the set as rendered can be understood as that between orderly stability within the boundary and chaotic instability beyond it. The many colourful renderings usually seen are generated by coloring points not in the set according to the degree of stability/instability. The axes of the plane divide the Mandelbrot set into 4 quadrants. More complex approaches to the dynamics of complexity include less familiar renderings of the Mandelbrot set in 3D, termed a mandelbulb. Also of relevance are variants termed multibrot sets, for which there are various images and animations (Animated morph of multibrots d = -7 to 7). It is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers. In contrast with the conventional orientation of that rendering (below left), a distinctive vertically oriented rendering, termed a Buddhabrot, has been developed -- so named because of its resemblance to a seated Buddha.

In the following exploration of the possibility of coherent comprehension of complexity, it is assumed that the complex plane can be rotated both on the real and on the imaginary axis. This results in the configuration of 3 mutually orthogonal planes. These can be understood as a feature of multiview projection, as shown in the central image below -- with the configuration framing 8 octants. Relevant to extensions of this argument, geometry recognizes the existence of an orthant (or hyperoctant) as the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.

Quadrants and Octants
Mandelbrot quadrants in complex plane	Octants in solid geometry	Octant sign convention	Octants with signs
		x y z I + + + II - + + III - - + IV + - + V + + - VI - + - VII - - - VIII + - -
Adapted from Wikipedia	Reproduced from Wikipedia	Reproduced from Wikipedia	Reproduced from Wolfram MathWorld

The following images are indicate of the complexity of the eightfold pattern -- and the degree of coherence suggested by the aesthetic nature of that framework.

Rotation of rendering of Mandelbrot set on axes of complex plane (using distinctive colours)
Minimal rotation on real axis	Minimal rotation on real and imaginary axes	Rotation of mutually orthogonal configuration
Animations developed using X3D-Edit

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