The $(n-1)$-dimensional simplex is the collection of probability measures on a set with $n$ points. Many applied situations result in simplex-valued data or in stochastic processes that have the simplex as their state space. In this paper we study a large class of simplex-valued diffusion processes that are constructed by first ``coordinatising'' the simplex with the points of a smooth hypersurface in such a way that several points on the hypersurface may correspond to a given point on the simplex, and then mapping forward the canonical Brownian motion on the hypersurface. For example, a particular instance of the Fleming-Viot process on $n$ points arises from Brownian motion on the $(n-1)$-dimensional sphere. The Brownian motion on the hypersurface has the normalised Riemannian volume as its equilibrium distribution. It is straightforward to compute the corresponding distribution on the simplex, and this provides a large class of interesting probability measures on the simplex.