Lack of relative monotonicity among various measures of trihedral angles (Q1926123)

While there is a clear (and somehow unique) idea of how to measure a plane angle, this is not the case when considering trihedral angles. In this paper five different measures of such magnitude are given. The paper then shows the fact that none of this measures are monotone with respect to any of the others. In other words, if \(f\) and \(g\) are two of those measures, there exist trihedral angles \(\alpha\), \(\beta\), \(\gamma\) and \(\theta\) such that \(f(\alpha)> f(\beta)\), \(g(\alpha)< g(\beta)\), \(f(\gamma)>f(\theta)\) and \(g(\gamma)>g(\theta)\).