  A fine mixed subdivision of a (d-1)-simplex T of size n gives rise to a
system of ${d \choose 2}$ permutations of [n] on the edges of T, and to a
collection of n unit (d-1)-simplices inside T. Which systems of permutations
and which collections of simplices arise in this way? The Spread Out Simplices
Conjecture of Ardila and Billey proposes an answer to the second question. We
propose and give evidence for an answer to the first question, the Acyclic
System Conjecture.
  We prove that the system of permutations of T determines the collection of
simplices of T. This establishes the Acyclic System Conjecture as a first step
towards proving the Spread Out Simplices Conjecture. We use this approach to
prove both conjectures for n=3 in arbitrary dimension.
