  Seven-point functions have two inequivalent topologies or channels. The comb
channel has been computed previously and here we compute scalar conformal
blocks in the extended snowflake channel in $d$ dimensions. Our computation
relies on the known action of the differential operator that sets up the
operator product expansion in embedding space. The scalar conformal blocks in
the extended snowflake channel are obtained as a power series expansion in the
conformal cross-ratios whose coefficients are a triple sum of the
hypergeometric type. This triple sum factorizes into a single sum and a double
sum. The single sum can be seen as originating from the comb channel and is
given in terms of a ${}_3F_2$-hypergeometric function, while the double sum
originates from the snowflake channel which corresponds to a Kamp\'e de
F\'eriet function. We verify that our results satisfy the symmetry properties
of the extended snowflake topology. Moreover, we check that the behavior of the
extended snowflake conformal blocks under several limits is consistent with
known results. Finally, we conjecture rules leading to a partial construction
of scalar $M$-point conformal blocks in arbitrary topologies.
