Geodesic polytopes in $\mathbb R^3$ can be used to construct "simple" triangulations of $\mathbb S^{2}$, the 2-sphere. They can be constructed, for example, by taking a regular octahedron subdividing each of its 8 facets into $m^2$ triangles and projecting each of those vertices to $\mathbb S^2$. Here $m$ is the number of subdivisions of its edges. Geodesic polytopes are simplicial and have a relatively simple construction, where both the coordinates of its vertices and their face enumeration are relatively easy to compute.
I was wondering if there are standard constructions for analogues on $\mathbb R^n$, that is for triangulating the $(n-1)$-sphere. One could for example start with the regular $n-dimensional$ cross polytope, subdivide each of its 2^n facets into $n-1$ simplexes and project the vertices of those simplexes to $\mathbb S^{n-1}$. A couple of difficulties arise, first one can not subdivide into regular simplexes anymore, so the concept of the face refinement (or ridge subdivision $m$) is hard to imagine. Moreover, the triangulation is also not intuitive anymore.
Is there a standard construction for Geodesic Polytopes in $n$-dimensions? I tried googling, but not much came out. I would appreciate any direction.