In multicomplex dynamics, the Tetrabrot is a 3D generalization of the Mandelbrot set[1]. Discovered by Dominic Rochon in 2000, it can be interpreted as a 3D slice ${\displaystyle {\mathcal {T}}^{2}(1,\mathbf {i_{1}} ,\mathbf {i_{2}} )}$ of the tricomplex Multibrot set ${\displaystyle {\mathcal {M}}_{3}^{2}}$[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[11], the algorithms use the tricomplex function ${\displaystyle f_{c}(\eta ):=\eta ^{p}+c}$ where ${\displaystyle \eta }$, ${\displaystyle c\in \mathbb {TC} }$ and ${\displaystyle p\geq 2}$ is an integer. Since ${\displaystyle c\in {\mathcal {T}}^{2}(1,\mathbf {i_{1}} ,\mathbf {i_{2}} )}$ if and only if ${\displaystyle |f_{c}^{m}(0)|\leq 2\,\,\forall m\geq 1}$. For a given number of iterations ${\displaystyle M}$, if the computations of ${\displaystyle f_{c}^{m}(0)}$ for ${\displaystyle m\in \left\lbrace 1,2,\ldots ,M\right\rbrace }$ is ${\displaystyle |f_{c}^{m}(0)|>2}$ 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.

Illustration of the Tetrabrot with the Divergence-Layer Algorithm

## Generalized Fatou-Julia theorem

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

${\displaystyle {\mathcal {K}}_{3,c}^{2}:=\left\lbrace \eta \in \mathbb {TC} \,:\,\left\lbrace f_{c}^{m}(\eta )\right\rbrace _{m=1}^{\infty }{\text{ is bounded}}\right\rbrace .}$

The basin of attraction at ${\displaystyle \infty }$ of ${\displaystyle f_{c}(\eta )=\eta ^{2}+c}$ is defined as ${\displaystyle A_{3,c}(\infty ):=\mathbb {TC} \smallsetminus {\mathcal {K}}_{3,c}^{2}}$, that is

${\displaystyle A_{3,c}(\infty )=\left\lbrace \eta \in \mathbb {TC} \,:\,f_{c}(\eta )\rightarrow \infty {\text{ as }}\eta \rightarrow \infty \right\rbrace }$

and the strong basin of attraction of ${\displaystyle \infty }$ of ${\displaystyle f_{c}}$ as

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

Illustration of the Fatou-Julia theorem for the Tetrabrot

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

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

In particular, ${\displaystyle {\mathcal {K}}_{3,c}^{2}}$ is connected if and only if ${\displaystyle c\in {\mathcal {M}}_{3}^{2}}$. 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[4] gave some algorithms for the generation and display of fractal shapes in 3D. For the first time, iteration with quaternions[5] appeared. Theoretical results have been treated for the quaternionic Mandelbrot set[6] [7] (see video) defined with quadratic polynomial in the quaternions of the form ${\displaystyle q^{2}+c}$.

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[8] and D. Rochon[9] obtained estimates for the lower and upper bounds of the distance from a point ${\displaystyle c}$ outside of the bicomplex Mandelbrot set ${\displaystyle {\mathcal {M}}_{2}^{2}}$ to ${\displaystyle {\mathcal {M}}_{2}^{2}}$ itself. Let ${\displaystyle c\not \in {\mathcal {M}}_{2}^{2}}$ and define

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

Then,

${\displaystyle d(c,{\mathcal {M}}_{2}^{2})={\sqrt {\frac {d(c_{\gamma _{1}},{\mathcal {M}}^{2})+d(c_{\overline {\gamma _{1}}},{\mathcal {M}}^{2})}{2}}}}$

where ${\displaystyle {\mathcal {M}}^{2}}$ is the standard Mandelbrot set.

The Mandelbrot set ${\displaystyle {\mathcal {M}}^{2}}$ with continuously colored environment

Using the Green function ${\displaystyle G:\mathbb {C} (\mathbf {i_{1}} )\smallsetminus {\mathcal {M}}^{2}\rightarrow \mathbb {C} (\mathbf {i_{1}} )\smallsetminus {\overline {B_{1}}}(0,1)}$ in the complex plane, where ${\displaystyle {\overline {B_{1}}}(0,1)}$ is the closed unit ball of ${\displaystyle \mathbb {C} (\mathbf {i_{1}} )\simeq \mathbb {C} }$, the distance is approximated in the following way[10]

${\displaystyle {\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 })|}}

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

Tetrabrot ray-traced

There exist also a generalization of the lower bound for ${\displaystyle d(c,{\mathcal {M}}_{2}^{2})}$ to the tricomplex Multibrot set of order ${\displaystyle p}$[11]. Some ressources and images can be found on the Aleph One's personal page[12]. There is also a video available on Youtube where specific regions of the Rochon's Tetrabrot are explored.

## References

1. ^ a b D. Rochon, "A Generalized Mandelbrot Set for Bicomplex Numbers", Fractals, 8(4):355-368, 2000. doi:10.1142/S0218348X0000041X
2. ^ a b V. Garrant-Pelletier and D. Rochon, "On a Generalized Fatou-Julia Theorem in Multicomplex Space", Fractals, 17(3):241-255, 2008. doi:10.1142/S0218348X03002075
3. ^ V. Garant-Pelleter, Ensemble de Mandlebrot et de Julia remplis classiques généralisés aux espaces multicomplexes et théorème de Fatou-Julia généralisé, Master's thesis, Université du Québec à Trois-Rivières, 2011.
4. ^ A. Norton, "Generation and Display of Geometric Fractals in 3-D", Computer Graphics, 16:61-67, 1982. doi:10.1145/965145.801263
5. ^ I. L. Kantor, Hypercomplex Numbers, Springer-Verlag, New-York, 1982.
6. ^ S. Bedding and K. Briggs, "Iteration of Quaternion Maps", Int. J. Bifur. Chaos Appl. Sci. Eng., 5:877-881, 1995. doi:10.1142/S0218127495000661
7. ^ J. Gomatam, J. Doyle, B. Steves and I. McFarlane, "Generalization of the Mandelbrot Set: Quaternionic Quadratic Maps", Chaos, Solitons & Fractals, 5:971-985, 1995. doi:10.1142/S0218127495000661
8. ^ É. Martineau, Bornes de la distance l'ensemble de Mandelbrot généralisé, Master's thesis, Université du Québec à Trois-Rivières, 2004.
9. ^ É. Martineau and D. Rochon, "On a Bicomplex Distance Estimation for the Tetrabrot", International Journal of Bifurcation and Chaos, 15(6):501-521, 2005. doi:10.1142/S0218127405013873
10. ^ J. C. Hart, D. J. Sandin and L. H. Kauffman, "Ray tracing deterministic 3-D fractals", Comput. Graph., 23:289-296, 1989.
11. ^ a b D. Rochon, "On a Tricomplex Distance Estimation for Generalized Multibrot Sets", CHAOS 2017. 3dfractals.com/docs/Chaos_2017.pdf
12. ^ http://aleph1.sourceforge.net/gallery/mandelbrot/