The Tetrabrot

In the previous section, we established a version of the Mandelbrot set in dimension four. Now, we want to give a version of the Mandelbrot set in dimension three using the definition for $\mathcal{M}_{2}$. The idea here is to preserve the Mandelbrot set inside $\mathcal{M}_{2}$. Then, if we restict the algorithm to the points of the form $a+bi_1+ci_2$ where $a,b,c\in\mathbb {R}$, we preserve the Mandelbrot set on two perpendicular complex planes and we stay in $\mathbb {R}^{3}$. This is the first argument to justify the following definition.

Definition 4   The ``Tetrabrot" is defined as follows: $\mathcal{T}=\{c=c_1+c_2i_2\in\mathbb {C}_{2}:Im(c_{2})=0 \mbox{ and }P_{c}^{\circ n}(0)\mbox{ is bounded }\forall n\in\mathbb {N}\}$.

We wish to give an illustration of the Tetrabrot in $\mathbb {R}^{3}$. The next result will give a manner to approach the Tetrabrot with the Euclidian norm in $\mathbb {R}^{4}$.

Theorem 3   $\mathcal{M}_{2}\subset\overline{D}(0,2)\subset\overline{{B}^{2}(0,2)}$ where $\overline{D}(0,2)=\overline{{B}^{1}(0,2)}\times_{e}\overline{{B}^{1}(0,2)}$. Moreover, the radius $2$ is the best possible in each case.

Proof. By Lemma 1, $\mathcal{M}_{2}=\mathcal{M}_{1}\times_{e}\mathcal{M}_{1}$. Moreover, $\overline{D}(0,2)=\overline{{B}^{1}(0,2)}\times_{e}\overline{{B}^{1}(0,2)}$ and $\mathcal{M}_{1}\subset\overline{{B}^{1}(0,2)}$ with a point of $\mathcal{M}_{1}$ which touches the boundary of the unit disc [1]. Then, $\mathcal{M}_{2}\subset\overline{D}(0,2)$ and the radius $2$ is the best possible. Finaly, it is proven in [8] that $\overline{D}(0,2)\subset\overline{{B}^{2}(0,2)}$ with points of $\overline{D}(0,2)$ which touche the boundary of $\overline{{B}^{2}(0,2)}$.$\Box$

Then, it is possible to compute the infinite divergence layers of the Tetrabrot from the number of iterations needed to have $\vert P_{c}^{\circ n}(0)\vert>2$. We have to remark here that each divergence layer will hide the others. For example, Fig. 2 is an illustration for the Tetrabrot of one of its divergence layers in correspondence with the divergence layer illustrated in Fig. 1.A for the Mandelbrot set. In fact, the Tetrabrot is inside Fig. 2. It is possible to see a part of the Tetrabrot (see Fig. 3) if we cut a piece of Fig. 2. In Fig. 3, the colours are an illustration of the other divergence layers. It is also possible to compute other divergence layers (see Figs. 4, 5 ,6 and 7). Figure 7 begins to be close to the set we wish to approach; then Fig. 7 with its cut plane gives certainly a good idea of the Tetrabrot.

To define the Tetrabrot, we have put the last coordinate in ``$j$" equal to zero. In fact it is possible to do the same thing if we fix the last coordinate equal to a number different from zero. However, if we do that, we lose the beautiful symmetry of the Tetrabrot. Figures 8 and 9 gives an illustration of this phenomenon for two different fixed ``$dj$" with $d\neq0$.

An interesting exploration of the Tetrabrot is now possible. For example Figure 10 is an enlargement of Fig. 7.A. It is also possible to be more specific. For example, Fig. 14 is an enlargement of 10.A and Figs. 11 and 12 are an enlargement of deep zones above the zone of Fig 7.A. Also, Fig. 13 is an enlargement of Fig 7.B. The colour in the background of Fig. 14 has been added to give a better 3-dimensional view. Each figure has been illustrated with a selected divergence layer and striations have been added to give an illustration of the ``level curves" of each figure.