Fractal simplices (Q2827424)

The \(n\)-simplex is a standard object in mathematics. It has a lot of interesting properties; for instance it is symmetric with its set of symmetries coinciding with the set of permutations of the set \(\{0,1,\ldots, n\}\). The author presents three \(n\)-simplex involved constructions which are higher-dimensional analogues of Sierpinski-like figures. He does so in hopes of aiding the study of persistent homology. The result of the first construction is an uncountably infinite figure in \(n\)-dimensional space that is Cantor-like in a way analogue to the Sierpinski triangle. To obtain this, one has to remove the open convex hull of the midpoints of the edges of the \(n\)-simplex and since the complement is a union of simplices, one then continues the removal recursively in each of the remaining sub-simplices. The second construction is a countable analogue which is obtained by playing the chaos game in the \(n\)-simplex. In this case we start by picking up an arbitrary \((n+1)\)-ary sequence; starting with the initial point we continue plotting points by moving half-again as much towards the next point in the sequence. The resulting plot converges to the aforementioned figure. Finally, the third construction entails coloring the multinomial coefficients black or white according to their parity; the colored infinite right simplex will approximate the Sierpinski simplex in a manner similar to the image of a generic chaos game.