The Minkowski tensors contain information about both the preferred direction and the amplitude of the anisotropy. The latter can be conveniently extracted by **scalar anisotropy indices**.

Here, we present some common indices in both 2D and 3D.

## Irreducible Minkowski Tensors (IMTs)

They provide the most systematic, sensitive and convenient characterization of interfacial anisotropy in a two-phase medium.

In particular, they characterize anisotropy w.r.t. different symmetries.

They are presented in detail in the main article: Anisotropy analysis by IMT

## Ratio of Eigenvalues

The Cartesian representation of the second-rank Minkowski tensors (with ) allows for a straightforward definition of anisotropy indices w.r.t. different geometrical properties (both translation invariant and covariant):

where and are the smallest and largest eigenvalues of .

indicates a “flat” body .

indicates an “isotropic” body , in the sense that it has a statistically identical mass distribution in any set of three orthogonal directions; this includes the sphere, but also regular polyhedra and the FCC, BCC and HCP Voronoi cells.

has be successfully applied to a variety of simulated and experimental systems.

In 2D, the index of the surface tensor is equivalent to the IMT .

## Isoperimetric Ratio

A truly classical index of anisotropy is the isoperimetric ratio .

In 2D:

In 3D:

In this sense, it is the dimensionless quotient of the volume and surface of an object. It is normalized so that for a ball.

Different shape information is contained in the isoperimetric ratio and the Minkowski anisotropy indices, as can be seen in the comparison of and for various geometric shapes: