Definition 34.1 (Barycenter of an affine simplex):

(i) The barycenter of a zero simplex, that is a point, is the zero simplex itself.

(ii) The barycenter of an affine $p$-simplex $[{\bf v}_1, {\bf v}_2,\dots, {\bf v}_{p+1}]$ is the point

$$\frac{1}{p+1}\Big( {\bf v}_1 + {\bf v}_2 + \dots + {\bf v}_{p+1} \Big) \eqno(34.4)$$

The barycenter of a one simplex is its midpoint and the barycenter of a two simplex is the centroid of the triangle determined by the vertices. Roughly speaking, the barycentric subdivision of a one simplex is obtained by subdividing the segment at its midpoint, or equivalently constructing the cone of each of the two endpoints with apex as the barycenter. To subdivide a two simplex, we first subdivide each of its three sides resulting in six one simplicies and taking the cone of each of the six pieces with apex as the barycenter of the two simplex. Figure below depicts these subdivisions. in More precisely the result of subdividing an affine $p-$simplex is a $p-$chain. The rough description above suggests an inductive definition.