  The analysis and visualization of tensor fields is a very challenging task.
Besides the cases of zeroth- and first-order tensors, most techniques focus on
symmetric second-order tensors. Only a few works concern totally symmetric
tensors of higher-order. Work on other tensors of higher-order than two is
exceptionally rare. We believe that one major reason for this gap is the lack
of knowledge about suitable tensor decompositions for the general higher-order
tensors. We focus here on three dimensions as most applications are concerned
with three-dimensional space. A lot of work on symmetric second-order tensors
uses the spectral decomposition. The work on totally symmetric higher-order
tensors deals frequently with a decomposition based on spherical harmonics.
These decompositions do not directly apply to general tensors of higher-order
in three dimensions. However, another option available is the deviatoric
decomposition for such tensors, splitting them into deviators. Together with
the multipole representation of deviators, it allows to describe any tensor in
three dimensions uniquely by a set of directions and non-negative scalars. The
specific appeal of this methodology is its general applicability, opening up a
potentially general route to tensor interpretation. The underlying concepts,
however, are not broadly understood in the engineering community. In this
article, we therefore gather information about this decomposition from a range
of literature sources. The goal is to collect and prepare the material for
further analysis and give other researchers the chance to work in this
direction. This article wants to stimulate the use of this decomposition and
the search for interpretation of this unique algebraic property. A first step
in this direction is given by a detailed analysis of the multipole
representation of symmetric second-order three-dimensional tensors.
