  Several powerful techniques for evaluating massless scalar Feynman diagrams
are developed, viz: the solution of recurrence relations to evaluate diagrams
with arbitrary numbers of loops in $n=4-2\omega$ dimensions; the discovery and
use of symmetry properties to restrict and compute Taylor series in $\omega$;
the reduction of triple sums over Chebyshev polynomials to products of Riemann
zeta functions; the exploitation of conformal invariance to avoid
four-dimensional Racah coefficients. As an example of the power of these
techniques we evaluate all of the 216 diagrams, with 5 loops or less, which
give finite contributions of order $1/k^2$ or $1/k^4$ to a propagator of
momentum $k$ in massless four-dimensional scalar field theories. Remarkably,
only 5 basic numbers are encountered: $\zeta(3)$, $\zeta(5)$, $\zeta(7)$,
$\zeta(9)$ and the value of the most symmetrical diagram, which is calculated
to 14 significant figures. It is conceivable that these are the only
irrationals appearing in 6-loop beta functions. En route to these results we
uncover and only partially explain many remarkable relations between diagrams.
