  We propose to imagine that every Riemannian metric on a surface is discrete
at the small scale, made of curves called walls. The length of a curve is its
number of wall crossings, and the area of the surface is the number of
crossings of the walls themselves. We show how to approximate a Riemannian (or
self-reverse Finsler) metric by a wallsystem.
  This work is motivated by Gromov's filling area conjecture (FAC) that the
hemisphere minimizes area among orientable Riemannian surfaces that fill a
circle isometrically. We introduce a discrete FAC: every square-celled surface
that fills isometrically a $2n$-cycle graph has at least $n(n-1)/2$ squares. We
prove that our discrete FAC is equivalent to the FAC for surfaces with
self-reverse metric.
  If the surface is a disk, the discrete FAC follows from Steinitz's algorithm
for transforming curves into pseudolines. This gives a new proof of the FAC for
disks with self-reverse metric. We also imitate Ivanov's proof of the same
fact, using discrete differential forms. And we prove that the FAC holds for
M\"obius bands with self-reverse metric. For this we use a combinatorial curve
shortening flow developed by de Graaf--Schrijver and Hass--Scott. With the same
method we prove the systolic inequality for Klein bottles with self-reverse
metric, conjectured by Sabourau--Yassine.
  Self-reverse metrics can be discretized using walls because every normed
plane satisfies Crofton's formula: the length of every segment equals the
symplectic measure of the set of lines that it crosses. Directed 2-dimensional
metrics have no Crofton formula, but can be discretized as well. Their
discretization is a triangulation where the length of each edge is 1 in one way
and 0 in the other, and the area of the surface is the number of triangles.
This structure is a simplicial set, dual to a plabic graph. The role of the
walls is played by Postnikov's strands.

Differential Geometry;Combinatorics;Geometric Topology;Metric Geometry