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algorithm help: making faces out of random verts on a sphere?

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I am trying to create a random solid by distributing a bunch of vertices randomly on a sphere. So far, so good. The hard part is creating faces out of the collection of random points (is this process called tesselation?). I am trying to make an algorithm to accomplish this by connecting closest vertices but it seems like it will rapidly become hopelessly complex. Can anybody give me some tips on how to accomplish this?

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would something like this work?

create first edge from any two neighboring verts

for each unchecked edge,
find closest unchecked vert.
add two edges (endpoints of this edge to new vert)to create a new face,
and add the two new edges to the list of unchecked edges.

but this wouldn''t guarantee a closed solid....ideas?

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