  We give a construction of MV-polytopes of a complex semisimple algebraic
group G in terms of the geometry of the Bott-Samelson variety and the affine
building. This is done by using the construction of dense subsets of MV-cycles
by Gaussent and Littelmann. They used LS-gallery to define subsets in the
Bott-Samelson variety that map to subsets of the affine Grassmannian, whose
closure are MV-cycles. Since points in the Bott-Samelson variety correspond to
galleries in the affine building one can look at the image of a point in such a
special subset under all retractions at infinity. We prove that these images
can be used to construct the corresponding MV-polytope in an explicit way, by
using the GGMS strata. Furthermore we give a combinatorial construction for
these images by using the crystal structure of LS-galleries and the action of
the ordinary Weyl group on the coweight lattice.
