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 of the tricomplex Multibrot set [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 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

The basin of attraction at of is defined as , that is

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]:

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.


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


where is the standard Mandelbrot set.

The Mandelbrot set with continuously colored environment

Using the Green function in the complex plane, where is the closed unit ball of , the distance is approximated in the following way[10]

and for large where and . 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 to the tricomplex Multibrot set of order [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.

See also


  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.
  12. ^

External links