Subdivide that tetrahedron, each face six ways to make poor Vinnies twenty four sider. 

Pick some points

Pick three points to make a triangle

Oops!

Friday the 13th passed uneventfully.

I can push the center of the initial triangle out along the cross product to find the center of my world.

Build a triangle mesh from three user picked points

Pick three points to make a triangle

Oops!

Build a 3D triangle surface mesh with vertices at least 10 px/1K km apart. We push out the edges at random from an initial triangle of three points at radius distance from the center of my hypothetical world, based on random size. detect concavities in the border edges of my world polyhedron (how?) and close them off with a bridging triangle and edge. Remove closed edges from the border and add new ones as we go along. Edges are two endpoints and the third vertex of a triangle with one or two closed edges and two or one new open edge. Edges are defined by endpoints, a third vertex of the triangle the edge belongs to. The midpoint and the vector through the midpoint from the third vertex are easy to derive. Or, edges are derived from adjacent vertices in the list of border points, checking for concavities to close off as we go. We get the vector from the midpoint of the edge by calculating the cross product of the two vectors from the center to the center to each endpoint. If this new point is too close to an existing border point, then we have detected a concavity and should close it off.

Derive a fourth coordinate from three points of user defined triangle

Pick three points to make a triangle

Oops!

Fill up a map rectangle with triangle fans of land and sea.