Let X be the space of all labeled tetrahedra in . In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285–312] we constructed a smooth symmetric compactification of X. In this article we show that the complement is a divisor with normal crossings, and we compute the cohomology ring .