CP1 structures are geometric structures modelled on the complex projective line, acted on by the projective group PSL(2,C). These structures are not as rigid as Riemmannian structure (like Euclidean, hyperbolic or spherical), nor as flexible as conformal structures. For example they still allow the notion of circles and therefore can be used to study circle packings. In this talk we show that all CP1 structures on the thrice punctured sphere -with elliptic holonomy- that are tame (i.e. the developing map extends continuously to the ends) can be constructed by elementary cutting and gluing (i.e. grafting) on simple triangular structures. The talk will be accessible to non-experts, with minimal background in topology.