2 1 Setting the Stage
we trace along a lacing, no three consecutive eyelets visited are contained in the
same column. The diagram in Figure 1.2 does not correspond to a lacing, as both
segments that end in one of the middle eyelets are contained in the same column
as this eyelet.
Figure 1.2. Not a lacing.
We call a segment of a lacing a vertical if its endpoints are both contained in
column A or both in column B. We call a segment a diagonal if it is not a vertical. A
diagonal is a horizontal if its endpoints have the same index, that is, are contained in
the same row. The vertical length of a segment is the nonnegative difference between
the indices of its endpoints. For example, the vertical length of a horizontal is 0.
A segment is an m-segment if its vertical length is m. For example, a 0-diagonal
is the same as a horizontal. The length of a lacing is the sum of the lengths of its
It is important to realize that a picture of a lacing can correspond to more than
one lacing. For example, consider Figure 1.3. The left- and right-hand diagrams
represent two different “loose” lacings that both contract to the middle diagram
when the shoelace is pulled taut. This means that, just by looking at the diagram
in the middle, we cannot be sure which lacing it represents. To avoid this kind of
ambiguity, we draw lacings loosely, whenever this is necessary.
Figure 1.3. Two “loose” lacings that contract to the same diagram.
1.1 Popular Lacings
Figure 1.4 shows one representative each from six popular families of lacings that
are actually used for various purposes. These are the crisscross, zigzag, star, bowtie,
serpent, and zigsag lacings.