Introduction

In 1982, A. Norton [7] gave some straightforward algorithms for the generation and display in 3-D of fractal shapes. For the first time, iteration with quaternions [6] appeared. Subsequently, theoretical results have been treated in [4] for the quaternionic Mandelbrot set defined with quadratic polynomial in the quaternions of the form $q^{2}+c$. However, in [2], S. Bedding and K. Briggs established that there is no interesting dynamics for this approach and it does not play any fundamental role analogous to that for the map $z^{2}+c$ in the complex plane. We note that another definition of a Mandelbrot set for the quaternions was introduced by J. Holbrook in [5]. This definition gives a Mandelbrot set in $\mathbb {R}^{3}$ which is not a slice of the quaternionic quadratic Mandelbrot set.

In this article, we use a commutative generalization of the complex numbers called bicomplex numbers ( [8], [9], [10], [11]) to give a new version of the Mandelbrot set in dimensions three and four. Moreover, we prove that our generalization in dimension four, noted $\mathcal{M}_{2}$, is a connected set. We also define the concept of ``filled-Julia" set for bicomplex numbers and we prove that a point is inside $\mathcal{M}_{2}$ if and only if the ``filled-Julia" set at that point is connected. These two results are perfectly analogous to the corresponding results in the complex plane.

Our generalization of the Mandelbrot set in dimension three is established from a slice of $\mathcal{M}_{2}$. We also give a graphics representation of our set, called the Tetrabrot, in $\mathbb {R}^{3}$ and we especially focus our attention on the infinite divergence layers to approach this set. Moreover, we give a graphics representation of the associated ``filled-Julia" set for points on the Tetrabrot and we note that shapes of certain ``filled-Julia" sets are reflected in the shape of the Tetrabrot near the corresponding points. This feature had also been remarked for the Mandelbrot set in the complex plane.

Finaly, we remark that the Tetrabrot could possibly be unconnected and we establish hypotheses about the geometry of the Mandelbrot set for which the Tetrabrot would be unconnected.