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The convex hull of n+1 affinely independent vertices of the unit n-cube Cn iscalled 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 ofsize 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 howto efficiently compute, by means of a computer program, a representative fromeach orbit of an acute 0/1-simplex under the action of the hyperoctahedralgroup Bn of symmetries of Cn. A side product of the investigations is a simplecode that computes the cycle index of Bn, which can in explicit form only befound in the literature for n < 7. Using the computed cycle indices incombination with Polya's theory of enumeration shows that acute 0/1-simplicesare extremely rare among all 0/1-simplices.  In the second part of the paper, we study the 0/1-matrices that represent theacute 0/1-simplices that were generated by our code from a mathematicalperspective. One of the patterns observed in the data involves unreduced upperHessenberg 0/1-matrices of size n x n, block-partitioned according to certaininteger compositions of n. These patterns will be fully explained using aso-called One Neighbor Theorem. Additionally, we are able to prove that thevolumes of the corresponding acute simplices are in one-to-one correspondencewith the part of Kepler's Tree of Fractions that enumerates the rationalsbetween 0 and 1. Another key ingredient in the proofs is the fact that theGramians of the unreduced upper Hessenberg matrices involved are strictlyultrametric matrices.

Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group