  We prove a conjecture of Goncharov, which says that any multiple
polylogarithm can be expressed via polylogarithms of depth at most half of the
weight. We give an explicit formula for this presentation, involving a
summation over trees that correspond to decompositions of a polygon into
quadrangles.
  Our second result is a formula for volume of hyperbolic orthoschemes,
generalizing the formula of Lobachevsky in dimension $3$ to an arbitrary
dimension. We show a surprising relation between two results, which comes from
the fact that hyperbolic orthoschemes are parametrized by configurations of
points on $\mathbb{P}^1.$ In particular, we derive both formulas from their
common generalization.
