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Tesselating a tetraheron and pushing the vertices out to unit length won't give me regular triangles. I tesselate by dividing all edges and creating 4 faces.
i don't quite get what you want to do. What you mean by "regular triangles"? Equilateral triangles?
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Since the distances from the origin to the original vertices and the distances to the centers of the edges are all the same i think my sphere should end up with uniform sized faces.
Where origin is? in center of tetrahedron?
What mean "sized"? Have equal area? "the distances to the centers of the edges" from what?
Do you set your vetrices after tesselation to be unit-length?
If you'll tesselate tetrahedron and then set all vertice vectors to be unit length, i'm almost sure you'll get a polyhedra that will have 4 bigger equilateral triangles and 12 smaller non-equilateral triangles, 16 triangles total.
What rules you want it to obey?
anyway, if you want to make polyhedron with vertices placed at equal distances from center, and composed of equal triangles, you can try
1:icosahedron, it have 20 equal faces(more than your 16-face polyhedra), and all edges is equal too, it'e regular polyhedron.
2:dodecahedron with additional vertice at center of each face, so each face is splitten into 5 triangles. Center vertices must be renormalized. So you'll have 60 equal triangles.(it will be slightly non-equilateral triangles)
edit: if by above rules you mean that thing about face edges, that 60-face thing made from dodecahedron probably will not work.