Hatcher-Algebraic-Topology · Problem 2.1.24

The barycentric subdivision of an $n$-simplex partitions it into smaller simplices, each corresponding to a total ordering of the coordinate indices. The key insight is that every point in the original simplex can be uniquely assigned to one of these smaller simplices based on which ordering its barycentric coordinates satisfy—the inequalities $t_{i_0} \leq t_{i_1} \leq \cdots \leq t_{i_n}$ act as a "sorting rule" that picks out exactly one permutation for each point, and these regions tile the simplex without overlap.

The barycentric subdivision of an $n$-simplex partitions it into smaller simplices, each corresponding to a total ordering of the coordinate indices. The key insight is that every point in the original simplex can be uniquely assigned to one of these smaller simplices based on which ordering its barycentric coordinates satisfy—the inequalities $t_{i_0} \leq t_{i_1} \leq \cdots \leq t_{i_n}$ act as a "sorting rule" that picks out exactly one permutation for each point, and these regions tile the simplex without overlap.