Setting the Stage
We start by describing the simple models of shoes and lacings of shoes that we will
be working with in the following.
A mathematical shoe consists of 2n eyelets which are the points of intersection of
two vertical lines and n equally spaced horizontal lines in the plane. In everything
that follows n ≥ 2. We fix the distance between the two vertical lines to be 1 and
call the distance h between two adjacent horizontal lines the stretch of the shoe;
see Figure 1.1. We call the set of all eyelets contained in one of the vertical lines
a column of eyelets and the set of two eyelets contained in a horizontal line a row
of eyelets. We label the left column with an A, the right column with a B, and the
rows of eyelets from 1 to n, proceeding from top to bottom. This induces a labelling
of the eyelets.
Figure 1.1. A mathematical shoe with stretch h.
An n-lacing of our mathematical shoe is a closed path in the plane consisting
of 2n line segments whose endpoints are the 2n eyelets of the shoe. Furthermore,
we require that, given any eyelet E, at least one of the two segments ending in it
is not contained in the same column as E. This condition ensures that every eyelet
genuinely contributes towards pulling the two sides of the shoe together or, less
formally, that lacings don’t have “gaps”. Virtually all lacings that are actually used
satisfy this condition. Note that our condition is equivalent to the following: As