My question is simple: How can I tessellate a sphere or a disk uniformly?

To just tessellate a sphere is an easy thing to implement using polar coordinates. However, this will have the effect of inreasing the LOD as you leave from the equator of the sphere to one of its poles. The same happens with circular disks. LOD will increase from the circumference of the disk to its center.

The reason for that is that the number of triangles per unit area will increase. So, I wonder if there is any algorithm that can tessellate a sphere or a disk down to a number of triangles that have an equal area.

There is a mathematical proof that shows that there is no solution when you try to proof that its impossible to tessalate a sphere uniformly by subdividing a tetrahedron. So therefore it must be possible. (Yes, mathematicians do this kinds of proofs.)

So use a tetrahedron as base and tessalate by aquiring the sphearical center of each tessalated triangle (which isnt the same as a linear center of a triangle.)