  Equifacetal simplices, all of whose codimension one faces are congruent to
one another, are studied. It is shown that the isometry group of such a simplex
acts transitively on its set of vertices, and, as an application, equifacetal
simplices are shown to have unique centers. It is conjectured that a simplex
with a unique center must be equifacetal. The notion of the combinatorial type
of an equifacetal simplex is introduced and analyzed, and all possible
combinatorial types of equifacetal simplices are constructed in even
dimensions.
