  In this paper we bring together tropical linear algebra and convex
3-dimensional bodies. We show how certain convex 3-dimensional bodies having 20
vertices and 12 facets can be encoded in a $4\times 4$ integer zero-diagonal
matrix $A$. A tropical tetrahedron is the set of points in $\R^3$ tropically
spanned by four given tropically non-coplanar points. It is a near-miss Johnson
solid. The coordinates of the points are arranged as the columns of a $4\times
4$ real matrix $A$ and the tetrahedron is denoted $\spann(A)$. We study
tropical tetrahedra which are convex and maximal, computing the extremals of
$\spann(A)$ and the length (tropical or Euclidean) of its edges. Then, we
classify convex maximal tropical tetrahedra, combinatorially. There are six
classes, up to symmetry and chirality. Only one class contains symmetric solids
and only one contains chiral ones. In the way, we show that the combinatorial
type of the regular dodecahedron does not occur here. We also prove that convex
maximal tropical tetrahedra are not vertex-transitive, in general. We give
families of examples, circulant matrices providing examples for two classes. A
crucial role is played by the $2\times 2$ minors of $A$.
