In multicomplex dynamics, the Tetrabrot^[1] is a 3D generalization of the Mandelbrot set. Discovered by Dominic Rochon in 2000, it can be interpreted as a 3D slice ${\mathcal{T}}^2(1,{\mathbf{i}}_{\mathbf{1}},{\mathbf{i}}_{\mathbf{2}})$ of the tricomplex Multibrot set ${\mathcal{M}}_3^2$.

Divergence-Layers Algorithm

There are different algorithms to generate pictures of the Tetrabrot. In the tricomplex space, the algorithms use the tricomplex function $f_c(\eta ):={\eta }^p+c$, where $\eta ,c\in \mathbb{TC}$ and $p\ge 2$ is an integer. The number $c\in {\mathcal{T}}^2(1,{\mathbf{i}}_{\mathbf{1}},{\mathbf{i}}_{\mathbf{2}})$ if and only if $|f_c^m(0)|\le 2$, for any integer $m\ge 1$. This condition means that the set of all numbers $f_c^m(0)$ should be bounded for every integer $m\ge 1$.

Since it is impossible to compute infinitely many iterations in a computer, we have to consider an approximation of the condition. Therefore, we fix a finite number of iterations to test, say $M$. The number $c$ belongs to the Tetrabrot if the numbers $f_c^m(0)$, for $m\in \{1,2,\dots ,M\}$, remain bounded by $2$. This is called the Divergence-Layer Algorithm. It is used to draw the Tetrabrot in the 3D space.

illustration of the Tetrabrot with the Divergence-Layer Algorithm

Generalized Fatou-Julia Theorem

The tricomplex filled-in Julia set of order $p=2$, for $c\in \mathbb{TC}$ is defined as $${\mathcal{K}}_{3,c}^2:=\{\eta \in \mathbb{TC}:\{f_c^m(\eta ){\}}_{m=1}^{\mathrm{\infty }}\text{ is bounded.}\}$$ The basin of attraction at $\mathrm{\infty }$ of $f_c(\eta )={\eta }^2+c$ is defined as $A_{3,c}(\mathrm{\infty }):=\mathbb{TC}\mathrm{\setminus }{\mathcal{K}}_{3,c}^2$, that is $$A_{3,c}(\mathrm{\infty })=\{\eta \in \mathbb{TC}:f_c^m(\eta )\to \mathrm{\infty }\text{ as }\eta \to \mathrm{\infty }\}$$ and the strong basin of attraction at $\mathrm{\infty }$ of $f_c$ as $$S{A}_{3,c}(\mathrm{\infty }):=(A_{c_{\gamma 1{\gamma }_3}}(\mathrm{\infty }){\times }_{{\gamma }_1}{A}_{c_{{\bar{\gamma }}_1{\gamma }_3}}(\mathrm{\infty })){\times }_{{\gamma }_3}(A_{c_{{\gamma }_1{\bar{\gamma }}_3}}(\mathrm{\infty }){\times }_{{\gamma }_1}{A}_{c_{{\bar{\gamma }}_1{\bar{\gamma }}_3}}(\mathrm{\infty })),$$ where $A_c(\mathrm{\infty })$ is the basin of attraction of $f_c$ at $\mathrm{\infty }$ for $c\in \mathbb{C}({\mathbf{i}}_{\mathbf{1}})$.

With these notations, the generalized Fatou-Julia Theorem for ${\mathcal{M}}_3^2$ is expressed in the following way^[2]:

• $0\in {\mathcal{K}}_{3,c}^2$ if and only if ${\mathcal{K}}_{3,c}^2$ is connected;
• $0\in S{A}_{3,c}(\mathrm{\infty })$ if and only if ${\mathcal{K}}_{3,c}^2$ is a Cantor set;
• $0\in A_{3,c}(\mathrm{\infty })\mathrm{\setminus }S{A}_{3,c}(\mathrm{\infty })$ if and only if ${\mathcal{K}}_{3,c}^2$ is disconnected but not totally.

Illustration of the Fatou-Julia Theorem for the Tetrabrot

Ray-Tracing

In 1982, A. Norton^[3] gave some algorithms for the generation and display of fractal shapes in 3D. For the first time, iteration with quaternions^[4] appeared. Theoretical results have been treated for the quaternionic Mandelbrot set^[5][6] (see video) defined with quadratic polynomial in the quaternions of the form $q^2+c$.

Quaternion Julia set with parameters $c=0.123+0.745i$ and with a cross-section in the $XY$ plane. The "Douady Rabbit" Julia set is visible in the cross section

In 2005, using bicomplex numbers, É. Martineau and D. Rochon^[7] obtained estimates for the lower and upper bounds of the distance from a point $c$ outside of the bicomplex Mandelbrot set ${\mathcal{M}}_2^2$ to ${\mathcal{M}}_2^2$ itself. Let $c\notin {\mathcal{M}}_2^2$ and define $$d(c,{\mathcal{M}}_2^2):=inf\{|w-c|:w\in {\mathcal{M}}_2^2.\}$$ Then, we have $$d(c,{\mathcal{M}}_2^2)=\sqrt{\frac{d(c_{{\gamma }_1},{\mathcal{M}}^2)+d(c_{{\bar{\gamma }}_1},{\mathcal{M}}^2)}{2}},$$ where ${\mathcal{M}}^2$ is the standard Mandelbrot set.

Using the Green function $G:\mathbb{C}({\mathbf{i}}_{\mathbf{1}})\mathrm{\setminus }{\mathcal{M}}^2\to \mathbb{C}({\mathbf{i}}_{\mathbf{1}})\mathrm{\setminus }{\bar{B}}_1(0,1)$ in the complex plane, where ${\bar{B}}_1(0,1)$ is the closed unit ball of $\mathbb{C}({\mathbf{i}}_{\mathbf{1}})\simeq \mathbb{C}$, the distance is approximated in the following way^[8]: $$\frac{|z_m|\ln |z_m|}{2|z_m{|}^{1/2^m}|z_m^{\prime }|}\approx \frac{\sinh G(c_{\gamma })}{2^{G(c_{\gamma })}|G^{\prime }(c_{\gamma })|}<d(c_{\gamma },{\mathcal{M}}^2),$$ for any $c_{\gamma }\in \mathbb{C}({\mathbf{i}}_{\mathbf{1}})\mathrm{\setminus }{\mathcal{M}}^2$ and for large $m$, where $z_m:=f_{c_{\gamma }}^m(0)$ and $z_m^{\prime }:=\frac{d}{dc}{f}_c(0){|}_{c=c_{\gamma }}$. This approximation gives a lower bound that can be used to ray-trace the Tetrabrot.

Tetrabrot ray-traced

There exists also a generalization of the lower bound for $d(c,{\mathcal{M}}_2^2)$ to the tricomplex Multibrot set of order $p$^[9]. Some ressources and images can be found on the Aleph One's personal page. There is also a video available on Youtube, where specific regions of the Rochon's Tetrabrot are explored.

References

1. ^D. Rochon, 'A Generalized Mandelbrot Set for Bicomplex Numbers', Fractals, 8(4):355-368, 2000.
2. ^V. Garant-Pelletier and D. Rochon, 'On a Generalized Fatou-Julia Theorem in Multicomplex Space', Fractals, 17(3):241-255, 2008.
3. ^A. Norton, 'Generation and Display of Geometric Fractals in 3-D', Computer Graphics, 16:61-67, 1982.
4. ^I. L. Kantor, Hypercomplex Numbers, Springer-Verlag, New-York, 1982.
5. ^S. Bedding and K. Briggs, 'Iteration of Quaternion Maps', Int. J. Bifur. Chaos Appl. Sci. Eng., 5:877-881, 1995
6. ^J. Gomatam, J. Doyle, B. Steves and I. McFarlane, 'Generalization of the Mandelbrot Set: Quaternionic Quadratic Maps', Chaos, Solitons & Fractals, 5:971-985, 1995.
7. ^É. Martineau and D. Rochon, 'On a Bicomplex Distance Estimation for the Tetrabrot', International Journal of Bifurcation and Chaos, 15(6):501-521, 2005.
8. ^J. C. Hart, D. J. Sandin and L. H. Kauffman, 'Ray tracing deterministic 3-D fractals', Comput. Graph., 23:289-296, 1989.
9. ^G. Brouillette, P.-O. Parisé & D. Rochon, 'Tricomplex Distance Estimation for Filled-in Julia Sets and Multibrot Sets', International Journal of Bifurcation and Chaos, 29, No. 6, 2019.