  In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3
dimensions was found, using the lattice algorithm PSLQ. Here we give a more
general formula, involving 3 distinct masses. A proof is devised, though it
cannot be accounted as a derivation; rather it certifies that an Ansatz found
by PSLQ satisfies a more easily derived pair of partial differential equations.
The result is similar to Schl\"afli's formula for the volume of a
bi-rectangular hyperbolic tetrahedron, revealing a novel connection between
3-loop diagrams and 1-loop boxes. We show that each reduces to a common basis:
volumes of ideal tetrahedra, corresponding to 1-loop massless triangle
diagrams. Ideal tetrahedra are also obtained when evaluating the volume
complementary to a hyperbolic knot. In the case that the knot is positive, and
hence implicated in field theory, ease of ideal reduction correlates with
likely appearance in counterterms. Volumes of knots relevant to the number
content of multi-loop diagrams are evaluated; as the loop number goes to
infinity, we obtain the hyperbolic volume of a simple 1-loop box.
