Pythagorean triples, rational angles, and space-filling simplices

The ancient Greeks posed and solved the problem of finding all right triangles with rational sidelengths. There are 4 natural nonEuclidean generalizations of this problem. We solve them all. The result is that the only rational-sided nonEuclidean triangle with one right angle is the isoceles spherical triangle with legs of length 45 and hypotenuse 60. We next ask which simplices have rational dihedral angles (measured in degrees). The solution is easy once connection is made to 1934 work of Coxeter. There are only a finite number of examples in 3-dimensional Euclidean space and only a countable number in n-space for n ≥ 4, which are nowhere dense in the space of simplices. But there are a dense and infinite set of examples if n = 2 or in nonEuclidean n-spaces for each n ≥ 2. In contrast, there are a continuum infinity of n-simplex shapes which tile n-space and are equidecomposable with n-parallelipipeds, as we show by explicit construction of infinite familes of of simplex tilers of n-space, for each n ≥ 4. (Some of these were previously known, while others are new.) There is a dense continuum infinity of n-simplex shapes with Dehn invariant 0. Along the way we prove that “Plouffe’s constant” and related angles are transcendental (Plouffe had not even known if they were rational). Although these four problems seem unrelated, they in fact are related.