  The convex hull of n+1 affinely independent vertices of the unit n-cube Cn is
called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse,
and acute if additionally none of them is right. In terms of linear algebra,
acute 0/1-simplices in Cn can be described by nonsingular 0/1-matrices P of
size n x n whose Gramians have an inverse that is strictly diagonally dominant,
with negative off-diagonal entries.
  The first part of this paper deals with giving a detailed description of how
to efficiently compute, by means of a computer program, a representative from
each orbit of an acute 0/1-simplex under the action of the hyperoctahedral
group Bn of symmetries of Cn. A side product of the investigations is a simple
code that computes the cycle index of Bn, which can in explicit form only be
found in the literature for n < 7. Using the computed cycle indices in
combination with Polya's theory of enumeration shows that acute 0/1-simplices
are extremely rare among all 0/1-simplices.
  In the second part of the paper, we study the 0/1-matrices that represent the
acute 0/1-simplices that were generated by our code from a mathematical
perspective. One of the patterns observed in the data involves unreduced upper
Hessenberg 0/1-matrices of size n x n, block-partitioned according to certain
integer compositions of n. These patterns will be fully explained using a
so-called One Neighbor Theorem. Additionally, we are able to prove that the
volumes of the corresponding acute simplices are in one-to-one correspondence
with the part of Kepler's Tree of Fractions that enumerates the rationals
between 0 and 1. Another key ingredient in the proofs is the fact that the
Gramians of the unreduced upper Hessenberg matrices involved are strictly
ultrametric matrices.
