Home Research Teaching Tetrabrot

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 \(\mathcal{T}^2(1, \mathbf{i_1}, \mathbf{i_2})\) of the tricomplex Multibrot set \(\mathcal{M}_3^2\).

Tetrabrot with Julia sets
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. The number $c \in \mathcal{T}^2 (1, \mathbf{i_1}, \mathbf{i_2})\) if and only if \(|f_c^m(0)| \leq 2\), for any integer \(m \geq 1\). For a given number of iterations \(M\), if the computations of \(f_c^m(0)\), for \(m \in \{1, 2, \ldots , M\}\) surface associated to different integers. This is called the Divergence-Layer Algorithm. It is used to draw the Tetrabrot in the 3D space.

Tetrabrot layered 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 $$ \mathcal{K}_{3, c}^2 := \{ \eta \in \mathbb{TC} \, : \, \{ f_c^m (\eta)\}_{m = 1}^\infty \text{ is bounded.} \} $$ The basin of attraction at \(\infty\) of \(f_c(\eta) = \eta^2 + c\) is defined as \(A_{3, c} (\infty) := \mathbb{TC} \backslash \mathcal{K}_{3, c}^2\), that is $$ A_{3, c} (\infty) = \{ \eta \in \mathbb{TC} \, : \, f_c^m (\eta ) \rightarrow \infty \text{ as } \eta \rightarrow \infty \} $$ and the strong basin of attraction of \(\infty\) of \(f_c\) as $$ SA_{3, c} (\infty ) := \Big( A_{c_{\gamma{1} \gamma_3}} (\infty) \times_{\gamma_1} A_{c_{\overline{\gamma}_1 \gamma_3}} (\infty ) \Big) \times_{\gamma_3} \Big( A_{c_{\gamma_1 \overline{\gamma}_3}} (\infty) \times_{\gamma_1} A_{c_{\overline{\gamma}_1 \overline{\gamma}_3}} (\infty) \Big) , $$ where \(A_c (\infty)\) is the basin of attraction of \(f_c\) at \(\infty\) for \(c \in \mathbb{C}(\mathbf{i_1})\).

Fatou-Julia Tetrabrot Illustration of the Fatou-Julia Theorem for the Tetrabrot


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 Douady rabbit 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 \(\mathcal{M}_2^2\) to \(\mathcal{M}_2^2\) itself. Let \(c \not\in \mathcal{M}_2^2\) and define $$ d(c, \mathcal{M}_2^2) := \inf \{ |w - c| \, : \, w \in \mathcal{M}_2^2 .\} $$ Then, we have $$ d(c, \mathcal{M}_2^2) = \sqrt{\frac{d(c_{\gamma_1}, \mathcal{M}^2) + d(c_{\overline{\gamma}_1}, \mathcal{M}^2)}{2}}, $$ where \(\mathcal{M}^2\) is the standard Mandelbrot set.

Using the Green function \(G : \mathbb{C}(\mathbf{i_1}) \backslash \mathcal{M}^2 \rightarrow \mathbb{C}(\mathbf{i_1}) \backslash \overline{B}_1 (0, 1)\) 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: $$ \frac{|z_m| \ln |z_m|}{2|z_m|^{1/2^m} |z_m'|} \approx \frac{\sinh G(c_\gamma)}{2^{G(c_\gamma)}|G'(c_\gamma)|} < d(c_\gamma, \mathcal{M}^2) , $$ for any \(c_\gamma \in \mathbb{C}(\mathbf{i_1}) \backslash \mathcal{M}^2\) 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.

Tetrabrot ray-traced 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 \(p\). 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.