  There is a remarkable well-known connection between the G$(4,n)$ cluster
algebra and $n$-particle amplitudes in $\mathcal{N}=4$ SYM theory. For $n \ge
8$ two long-standing open questions have been to find a mathematically natural
way to identify a finite list of amplitude symbol letters from among the
infinitely many cluster variables, and to find an explanation for certain
algebraic functions, such as the square roots of four-mass-box type, that are
expected to appear in symbols but are not cluster variables. In this letter we
use the notion of "stringy canonical forms" to construct polytopal realizations
of certain compactifications of (the positive part of) the configuration space
Conf${}_n(\mathbb{P}^{k-1}) \cong {\rm G}(k,n)/T$ that are manifestly finite
for all $k$ and $n$. Some facets of these polytopes are naturally associated to
cluster variables, while others are naturally associated to algebraic functions
constructed from Lusztig's canonical basis. For $(k,n) = (4,8)$ the latter
include precisely the expected square roots, revealing them to be related to
certain "overpositive" functions of the kinematical invariants.
