  We present for the first time fully analytic results for multi-loop
equal-mass ice cone graphs in two dimensions. By analysing the leading
singularities of these integrals, we find that the maximal cuts in two
dimensions can be organised into two copies of the same periods that describe
the Calabi-Yau varieties for the equal-mass banana integrals. We obtain a
conjectural basis of master integrals at an arbitrary number of loops, and we
solve the system of differential equations satisfied by the master integrals in
terms of the same class of iterated integrals that have appeared earlier in the
context of equal-mass banana integrals. We then go on and show that, when
expressed in terms of the canonical coordinate on the moduli space, our results
can naturally be written as iterated integrals involving the geometrical
invariants of the Calabi-Yau varieties. Our results indicate how the concept of
pure functions and transcendental weight can be extended to the case of
Calabi-Yau varieties. Finally, we also obtain a novel representation of the
periods of the Calabi-Yau varieties in terms of the same class of iterated
integrals, and we show that the well-known quadratic relations among the
periods reduce to simple shuffle relations among these iterated integrals.
