The Mandelbox, an artistic and geometric journey
The Mandelbox is a mathematical object that was first presented by Tom Lowe (aka Tglad) early in 2010. It can represent properties of the Mandelbrot set in dimensions other than the two of the Mandelbrot set. In contrast to Quaternion Fractals or the Mandlbulb, which follow a formal mathematical path with the Mandelbrot iteration z=z²+c, the Mandelbox follows a purely geometrical approach. On the one hand, this allows representations in any number of dimensions. On the other hand, the shape is relatively independent from the shapes known from Mandelbrot and Julia sets.
Since up to now only graphical applications are known, higher-dimensional variants hardly play a role. The vast majority of the images of the Mandelbox are three dimensional variants. (There are a few three-dimensional representations of a 4d Mandelbox, but only a few)
Fundamental transformation of the Mandelbrottian iteration (z=z²+c) is the squaring of a complex number represented in z². In the vectorial view this corresponds to a rotational extension of the position vector of a point. In other words, this is a circular transformation with subsequent scaling. In the Mandelbox, a spherical transformation (the so-called BallFold) with subsequent scaling is performed. This is already the whole trick. In addition, there is a BoxFold for the shape and - as with Mandelbrot - the addition of a constant c.
Please consider that this reflects a pointed, purely geometric view. The algebraic view may see the Mandelbox with a completely different emphasis. I have deliberately chosen this view, because I have a good geometric intuition, sufficient expertise and as a CAD professional a lot of experience with geometric relations. This is in no way meant to diminish the fantastic work of those who have formulated these calculations and translated them into valid and fast algorithms.
In most common 3d fractal software - Mandelbulber and Mandelbulb3d - the default starting Parameter at a scale of two
An image of 2011, from the early days of the Mandelbox. The Mandelbox can also be scaled negatively, which corresponds to a rotation of 180°. With additional rotations of about 0°-15°, images like this one are created. One of the first parameters, I converted to Mandelbulb3d. With help of Jens Dierks (aka Jesse) - the Programmer of M3d
Published in: Benoit Mandelbrot: A life in many dimensions
An unrotated Mandelbox at a scale of -2,5. Shape was developed in Mandelbulber, ported to Mandelbulb3d and rendered in M3d MC renderer