The diagrammatic theory of knotted surfaces in $4$-space started with Yajima's work in the $1960$'s. He showed that if a knotted sphere has a projection into $3$-space without triple points then it is a ribbon sphere. In the $1980$'s Giller developed the diagrammatic theory further and Roseman gave seven fundamental moves those are analogous to the Reidemeister moves. More recently, Kamada developed the braid form of a knotted surface following Viro and Rudolph. A series of works due to Carter and Saito in the $1990$'s are among the most important foundations of the theory.

The triple point number of a knotted surface is the minimal number of triple points among all projections of the surface. This notion is an analogue of the crossing number of a classical knot. There were no examples of non-ribbon spheres whose triple point numbers were concretely determined. We prove that Zeeman's $2$- and $3$-twist-spun trefoils have the triple point numbers $4$ and $6$ respectively. Furthermore, infinitely many examples of knotted spheres with triple point number $4$ and $6$ can be constructed. In the proof we use the quandle cocycle invariants defined by Carter, Jelsovsky, Kamada, Langford, and Saito in $1999$.