TITLE: About a Delzant polytope. (In particular dodecahedron)
QUESTION [19 upvotes]: Hi. I have a  question. 
Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ form an element of $SL(n,\mathbb{Z})$, where $n$ is a dimension of $P$.
(If you wonder why this condition is called smooth, See Fulton. Introduction to toric variety chap I)
My question is as follow. 
Can dodecahedron be the Delzant polytope?
I mean, is there a symplectic toric manifold whose moment map image is combinatorially equivalent to a dodecahedron?
Delzant's classfication theorem of compact symplectic toric manifold is surely very strong. But I think it is very hard to check whether the given polytope (having many faces)
is of Delzant type or not. If you know any reference of give me any comment, I really appriciate for your help. 
Thank you.

REPLY [7 votes]: Here is the answer to your question:
Elisa Prato, Symplectic Toric Geometry and the Regular Dodecahedron (2015)
Sorry it took so long. Hope you find this helpful. Another article covering all the platonic solids, rational or not, is on its way.