  We address the appearance of algebraic singularities in the symbol alphabet
of scattering amplitudes in the context of planar $\mathcal{N}=4$ super
Yang-Mills theory. We argue that connections between cluster algebras and
tropical geometry provide a natural language for postulating a finite alphabet
for scattering amplitudes beyond six and seven points where the corresponding
Grassmannian cluster algebras are finite. As well as generating natural finite
sets of letters, the tropical fans we discuss provide letters containing square
roots. Remarkably, the minimal fan we consider provides all the square root
letters recently discovered in an explicit two-loop eight-point NMHV
calculation.
