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The transformation is non-uniform scale - it does not preserve angles. The only reason the angles for the other shape were preserved was because the scaling was in the same direction as the edge. To get the result that you want, you either have to scale by the same amount in each direction, or just rescale the "viewing window" and not the shapes.


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Build a hexahedron from 6 triangles(!), setting its 2 vertices at the poles of the sphere, and the other 3 vertices equally around the equator. Then iteratively or recursively subdivide all triangles, projecting the new vertices at the sphere surface (!). Triangle subdivision can be done by replacing an original triangle by 4 new ones, adding new vertices in ...


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