function to a polytope composed by its Feynman diagrams and poles, making the map in more intuitive as now we discuss.

Polytopes from Cayley function
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We have seen that CHY formula for $C_n^2$ produces a set of Feynman diagrams, $T(C_n)$, each with n-3 poles; two Feynman diagrams can share $n-4$ poles, and such objects with $n-4$ poles can start to share $n-5$ poles, and so on, until we reach the set of all poles $P(C_n)$. Combinatorically, they can be represented as a polytope in $n-3$ dimensions. In this section, we describe the construction of such polytopes, and especially give a direct map from Cayely functions or labelled trees to these polytopes.

[ **Polytope of Feynman diagrams :**]{} Each vertex of this polytope corresponds to a Feynman diagram which is a set of $(n-3)$ compatible poles. Each edge corresponds to a set of $(n-4)$ compatible poles. Two vertices are connected by an edge iff their Feynman diagrams share $n-4$ poles, which correspond to the intersection of the two diagrams ( so there are always $n-3$ edges stretching out from each vertex ). Similarly, a dimension-$r$ face corresponds to a set of $n-3-r$ compatible poles, which is the intersection of those 