  We present an exposition of a remarkable example attributed to Frederick
Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain
definitions in the calculus of variations.
  The Almgren-Federer example, besides its intended goal of illustrating subtle
aspects of geometric measure theory, is also a problem in the theory of
geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and
Federer from the point of view of geodesics.
  In the language of geodesics, Almgren-Federer example constructs metrics in
$\mathbb{S}^1\times \mathbb{S}^2$, with the property that none of the Tonelli
geodesics (geodesics which minimize the length in a homotopy class) are Class-A
minimizers in the sense of Morse (any finite length segment in the universal
cover minimizes the length between the end points; this is also sometimes given
other names). In other words, even if a curve is a minimizer of length among
all the curves homotopic to it, by repeating it enough times, we get a closed
curve which does not minimize in its homotopy class.
  In that respect, the example is more dramatic than a better known example due
to Hedlund of a metric in $\mathbb{T}^3$ for which only 3 Tonelli minimizers
(and their multiples) are Class-A minimizers.
  For dynamics, the example also illustrates different definitions of
`integrable' and clarifies the relation between minimization and hyperbolicity
and its interaction with topology.
