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
where
,
and
is an integer. Since
if and only if
. For a given number of iterations
, if the computations of
for
is
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
for
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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/804ba823398827c36a66fef78a57bbe4ebe3554f)
The basin of attraction at
of
is defined as
, 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 }](https://web.archive.org/web/20171117200453im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/5100552f727b5cf1085dc9c5463ec49a4af11c58)
and the strong basin of attraction of
of
as
where
is the basin of attraction of
at
for
.
Illustration of the Fatou-Julia theorem for the Tetrabrot
With these notations, the generalized Fatou-Julia theorem for
is expressed in the following way[2][3]:
if and only if
is connected;
if and only if
is a Cantor set;
if and only if
is disconnected but not totally.
In particular,
is connected if and only if
. 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
.
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
outside of the bicomplex Mandelbrot set
to
itself. Let
and define
![{\displaystyle d(c,{\mathcal {M}}_{2}^{2}):=\inf \left\lbrace |w-c|\,:\,w\in {\mathcal {M}}_{2}^{2}\right\rbrace .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a52f3daa89d7ca146d15dcd3f84f83beff59ded)
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4a31733d2d826f40dd2415dc3de05aa33da601)
where
is the standard Mandelbrot set.