  We present a generalization of standard Turing machines based on allowing
unusual tapes. We present a set of reasonable constraints on tape geometry and
classify all tapes conforming to these constraints. Surprisingly, this
generalization does not lead to yet another equivalent formulation of the
notion of computable function. Rather, it gives an alternative definition of
the recursively enumerable Turing degrees that does not rely on oracles. The
definitions give rise to a number of questions about computable paths inside
Cayley graphs of finitely generated groups, and several of these questions are
answered.
