Fractal simplices

Abstract

There are three constructions of which I know that yield higher dimensional analogues of Sierpinski’s triangle. The most obvious is to remove the open convex hull of the midpoints of the edges of the [Formula: see text]-simplex. The complement is a union of simplices. Continue the removal recursively in each of the remaining sub-simplices. The result is an uncountably infinite figure in [Formula: see text]-dimensional space that is Cantor-like in a manner analogous to the Sierpinski triangle. A countable analogue is obtained by means of playing the chaos game in the [Formula: see text]-simplex. In this “game” one chooses a random [Formula: see text]-ary sequence; starting from the initial point (that is identified with a vertex of the simplex), one continues to plot points by moving half-again as much towards the next point in the sequence. The resulting plot converges to the figure described above. Similarly, coloring the multinomial coefficients black or white according to their parity results in a similar figure, when the [Formula: see text]-dimensional analogue of the Pascal triangle is rescaled and embedded in space.