  We examine the remarkable connection, first discovered by Beukers, Kolk and
Calabi, between $\zeta(2n)$, the value of the Riemann zeta-function at an even
positive integer, and the volume of some $2n$-dimensional polytope. It can be
shown that this volume is equal to the trace of a compact self-adjoint
operator. We provide an explicit expression for the kernel of this operator in
terms of Euler polynomials. This explicit expression makes it easy to calculate
the volume of the polytope and hence $\zeta(2n)$. In the case of odd positive
integers, the expression for the kernel enables us to rediscover an integral
representation for $\zeta(2n+1)$, obtained by a different method by
Cvijovi\'{c} and Klinowski. Finally, we indicate that the origin of the
miraculous Beukers-Kolk-Calabi change of variables in the multidimensional
integral, which is at the heart of this circle of ideas, can be traced to the
amoeba associated with the certain Laurent polynomial. The paper is dedicated
to the memory of Vladimir Arnold (1937-2010).
