  Cluster categories and cluster algebras encode two dimensional structures.
For instance, the Auslander--Reiten quiver of a cluster category can be drawn
on a surface, and there is a class of cluster algebras determined by surfaces
with marked points.
  Cluster characters are maps from cluster categories (and more general
triangulated categories) to cluster algebras. They have a tropical shadow in
the form of so-called tropical friezes, which are maps from cluster categories
(and more general triangulated categories) to the integers.
  This paper will define higher dimensional tropical friezes. One of the
motivations is the higher dimensional cluster categories of Oppermann and
Thomas, which encode $( d+1 )$-dimensional structures for an integer $d
\geqslant 1$. They are $( d+2 )$-angulated categories, which belong to the
subject of higher homological algebra.
  We will define higher dimensional tropical friezes as maps from higher
cluster categories (and more general $( d+2 )$-angulated categories) to the
integers. Following Palu, we will define a notion of $( d+2 )$-angulated index,
establish some of its properties, and use it to construct higher dimensional
tropical friezes.
