Floret Tessellation of a Sphere

Solution 1:

Start with an icosahedron, and replace each triangle with the interior of this picture, which is to say, pick any two vertices of degree six in that picture (the centers of the florets), complete the equilateral triangle, and map the interior points linearly onto the triangle of the icosahedron. As a final step, push all the new vertices out onto the bounding sphere to make it more spherical-looking. The process is very similar to making a geodesic dome model, except you are using a more interesting tiling than the standard map. If you find it easier, you can generate the floret tiling as the dual of the snub hexagonal tiling, which is to say, connect the centers of the faces of this tiling to get the floret tiling. The construction is closely related to the Pentagonal hexecontahedron, which can be viewed as the most basic version of this, where you choose the smallest possible triangle in the tiling and hence only get pentagonal pieces.