This page is linked from here; go there for some background. More explanation about the construction of this object is also here.

Nice balancing job in the third image! (You know, of course, that this is only possible during an equinox.)

There are three ways to add a tetrahedron to the augmented tetrahedron (7V2) to get a new (8-vertex) deltahedron. Which ones among the following 14 are they? (Source for image: this paper by N. Tsuruta, J. Mitani, Y. Kanamori, Y. Fukui (2015), Random Realization of Polyhedral Graphs as Deltahedra, Journal for Geometry and Graphics, Volume 19 (2015), No. 2, 227–236.)