This paper is about -triangles, which are the simplest nontrivial examples of -polytopes: convex hulls of a subset of vertices of the unit -cube . We consider the subclasses of right -triangles, and acute -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of .
The convex hull of n + 1 affinely independent vertices of the unit n-cube In 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 In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description...
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