The paper by Gabe Brisson and Cliff Reiter [1] and also the text [2] describe a method for visualizing generalizations of the Sierpinski Triangle in any dimension. The generalization of the Sierpinski Triangle to three dimensions is a Sierpinski Tetrahedron. This animation shows the four dimensional version rotated so that its structure and symmetry is more apparent. The structure of these fractals are based on a stroke based construction. Note the four-fold branching and the many Sierpinski Triangles appearing in planes throughout this example.

[1] Gabriel F. Brisson and Clifford A. Reiter, "Sierpinski Fractals from Words in High Dimension", Chaos, Solitons & Fractals, 5 11 (1995) 2191-2200.
[2] Clifford A. Reiter, Fractals, Visualization and J, Iverson Software, Inc., Toronto (1995).