Conclusion

The last theorem is a good indication that the conjecture is true because the hypothesis about the Mandelbrot set can be approximately confirmed by computers with a high level of precision. To confirm that the conjecture is true, we have two choices: to demonstrate theoretically the hypothesis about the geometry of the Mandelbrot set or to prove more directly that the Tetrabrot is unconnected. If the conjecture is proven to be true, a new question could be to know the cardinality of the family of the connected components. Also, it could be interesting to know whether the ``filled-Julia" set associated with points on an unconnected piece of the Tetrabrot have some specific proporties such as to be also unconnected. Finaly, another pertinent question could be to know the local fractal dimension of the boundary of the Tetrabrot.