I've been reading Beck's polytope textbook, and have been supplementing that with some of Ionascu's work regarding Ehrhart polynomials, which encode "the relationship between the volume of a[n integral] polytope and the number of integer points the polytope contains."
However, while reading Ionascu's "Lattice Platonic Solids and their Ehrhart Polynomial" (arXiv PDF), specifically section 4 ("Regular Tetrahedrons and regular Octahedrons"), I got pretty confused.
Regarding the tetrahedron and octahedron, he provides a lucid treatment for computing the coefficients of the $t^3$ and $t^2$ terms. However, he derives a relation between the coefficients in the tetrahedron and octahedron cases for the coefficients of $t$ and leaves it there, saying that Dedekind sums are needed to further his discussion. I don't really see how to do anything about the third coefficient. Could someone please expand on his thoughts at the end of the paper?
And even further, can someone show how to find the coefficient of $t$ for $T_3$ or any one of his examples?
Indeed, Ionascu doesn't compute the coefficient of $t$. He was probably hoping for easier machinery to compute the coefficient.
Pommersheim gave a formula for the number of lattice points of lattice tetrahedron given in terms of the tetrahderon's volume, lattice areas of its facets, lattice lengths of its edges, and functions of the dihedral angles computed via Dedekind sums. Hence, the coefficient of $t$ in the Ehrhart polynomial is expressed using Dedekind sums. Pommersheim gives an example for the Mordell tetrahedron.
Diaz, Le, and Robins use a Fourier transform of a polytope's indicator function, the Poisson summation formula, and the polytope's face poset to come up with an expression for the polytope's solid-angle polynomial. Machado and Robins show how to recover the coefficients of the Ehrhart polynomial from the solid-angle polynomial and give two examples of how this is done.