In multicomplex dynamics, the Tetrabrot is a 3D generalization of the Mandelbrot set. Discovered by Dominic Rochon in 2000, it can be interpreted as a 3D slice ${\mathcal {T}}^{2}(1,\mathbf {i_{1}} ,\mathbf {i_{2}} )$ of the tricomplex Multibrot set ${\mathcal {M}}_{3}^{2}$ . Illustration of filled-in Julia sets related to the Tetrabrot

## 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 $p\geq 2$ is an integer. Since $c\in {\mathcal {T}}^{2}(1,\mathbf {i_{1}} ,\mathbf {i_{2}} )$ if and only if $|f_{c}^{m}(0)|\leq 2\,\,\forall m\geq 1$ . For a given number of iterations $M$ , if the computations of $f_{c}^{m}(0)$ for $m\in \left\lbrace 1,2,\ldots ,M\right\rbrace$ is $|f_{c}^{m}(0)|>2$ a level surface can be associated to this integer. In this way, it is possible to draw a different level surface associated to different integers. This is called the Divergence-Layer Algorithm. It is used to draw the Tetrabrot in the 3D space.

## Generalized Fatou-Julia theorem

The tricomplex filled-in Julia set of order $p=2$ for $c\in \mathbb {TC}$ is defined as

${\mathcal {K}}_{3,c}^{2}:=\left\lbrace \eta \in \mathbb {TC} \,:\,\left\lbrace f_{c}^{m}(\eta )\right\rbrace _{m=1}^{\infty }{\text{ is bounded}}\right\rbrace .$ The basin of attraction at $\infty$ of $f_{c}(\eta )=\eta ^{2}+c$ is defined as $A_{3,c}(\infty ):=\mathbb {TC} \smallsetminus {\mathcal {K}}_{3,c}^{2}$ , that is

$A_{3,c}(\infty )=\left\lbrace \eta \in \mathbb {TC} \,:\,f_{c}(\eta )\rightarrow \infty {\text{ as }}\eta \rightarrow \infty \right\rbrace$ and the strong basin of attraction of $\infty$ of $f_{c}$ as

$SA_{3,c}(\infty ):=\left[A_{c_{\gamma _{1}\gamma _{3}}}(\infty )\times _{\gamma _{1}}A_{c_{{\overline {\gamma _{1}}}\gamma _{3}}}(\infty )\right]\times _{\gamma _{3}}\left[A_{c_{\gamma _{1}{\overline {\gamma _{3}}}}}(\infty )\times _{\gamma _{1}}A_{c_{{\overline {\gamma _{1}}}{\overline {\gamma _{3}}}}}(\infty )\right]$ where $A_{c}(\infty )$ is the basin of attraction of $f_{c}$ at $\infty$ for $c\in \mathbb {C} (\mathbf {i_{1}} )$ .

With these notations, the generalized Fatou-Julia theorem for ${\mathcal {M}}_{3}^{2}$ is expressed in the following way:

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

In particular, ${\mathcal {K}}_{3,c}^{2}$ is connected if and only if $c\in {\mathcal {M}}_{3}^{2}$ . For each statements, a specific color can be assigned to a specific case to obtain some information on the topology of the set.

## Ray-tracing

In 1982, A. Norton gave some algorithms for the generation and display of fractal shapes in 3D. For the first time, iteration with quaternions appeared. Theoretical results have been treated for the quaternionic Mandelbrot set  (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 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 $c\not \in {\mathcal {M}}_{2}^{2}$ and define

$d(c,{\mathcal {M}}_{2}^{2}):=\inf \left\lbrace |w-c|\,:\,w\in {\mathcal {M}}_{2}^{2}\right\rbrace .$ Then,

$d(c,{\mathcal {M}}_{2}^{2})={\sqrt {\frac {d(c_{\gamma _{1}},{\mathcal {M}}^{2})+d(c_{\overline {\gamma _{1}}},{\mathcal {M}}^{2})}{2}}}$ where ${\mathcal {M}}^{2}$ is the standard Mandelbrot set. The Mandelbrot set ${\mathcal {M}}^{2}$ with continuously colored environment

Using the Green function $G:\mathbb {C} (\mathbf {i_{1}} )\smallsetminus {\mathcal {M}}^{2}\rightarrow \mathbb {C} (\mathbf {i_{1}} )\smallsetminus {\overline {B_{1}}}(0,1)$ in the complex plane, where ${\overline {B_{1}}}(0,1)$ is the closed unit ball of $\mathbb {C} (\mathbf {i_{1}} )\simeq \mathbb {C}$ , the distance is approximated in the following way

${\frac {|z_{m}|\ln |z_{m}|}{2|z_{m}|^{1/2^{m}}|z_{m}'|}}\approx {\frac {\sinh G(c_{\gamma })}{2^{G(c_{\gamma })}|G'(c_{\gamma })|}} $\forall c_{\gamma }\in \mathbb {C} (\mathbf {i_{1}} )\smallsetminus {\mathcal {M}}^{2}$ and for large $m$ where $z_{m}:=f_{c_{\gamma }}^{m}(0)$ and $z_{m}':=\left.{\frac {d}{dc}}f_{c}(0)\right|_{c=c_{\gamma }}$ . This approximation gives a lower bound that can be used to ray-trace the Tetrabrot.

There exist also a generalization of the lower bound for $d(c,{\mathcal {M}}_{2}^{2})$ to the tricomplex Multibrot set of order $p$ . 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.