An Intrinsically Three-Dimensional Fractal

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits selfsimilar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the twodimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Twoand three-dimensional renderings are presented to explore some of the features of this set.