An unshellable triangulation of a tetrahedron

A triangulation if of a tetrahedron T is shellable if the tetrahedra Ku • • é , Kn of K can be so ordered that KJUK*-+iVJ • • • UüCn is homeomorphic to Tfor i=l, • • • , n. Sanderson [Proc. Amer. Math. Soc. vol. 8 (1957) p. 917] has shown that, if if is a Euclidean triangulation of a tetrahedron then there is a subdivision K' of K which is shellable; and he raises the question of the existence of a Euclidean triangulation of a tetrahedron which is unshellable. Such a triangulation will be described here. Let T be a tetrahedron each of whose edges has length 1. We will describe a nontrivial Euclidean triangulation K of T such that, if R is any tetrahedron of K, then the closure of (T — R) is not homeomorphic to T. I. Construction of K: Let Xi, X2, X$, and X* be the vertices of T. The possible values for the letters i and j are 1,2,3, and 4 and addition involving i or j will be modulo 4. For each i, let F* denote the face of T opposite Xit and let Ui be the midpoint of the interval X,-X;+2. Observe that U\ = U% and