This article discusses some rather technical properties of pair traversal diagrams diagrams as well as thread diagrams.

The text input files are machine-friendly versions of the basic pair traversal diagrams. The first line in a text file specifies the dimensions of a matrix, the next lines are rows of a matrix. Each matrix element represents a node in a pair traversal diagram.

Table of Contents Nodes / Tuples Types of Tiles Matrix for a diamond tiles mapped on a checkerboard Flanders

The code calls the matrix elements tuples. Each element in a tuple corresponds with a neighboring node. A positive number means something incoming, a negative number something outgoing, a zero means no connection. A one is just a pair, two a plait; see this discussion. The tuple for the red dot in the image is (1,0,0,1,0,0,-1,-1)

The templates for the basic pair traversals used to have shorter tuples: a3 and a7 were omitted. With only six adjacent nodes, a2-a4 not negative and a6-a8 not positive and omitting plaits, we can identify 20 possible tuples for the basic diagrams. With a tuple of 8 positions and plaits there are [NoInterleaved] possibilities.

Tiles can have different shapes and arrangements, to name a few: diamonds, bricks and checkerboards. The templates of DiBL use checkerboard tiles to create the main diagram. When a pair traversal pattern is defined with another type of tiles, the checkerboard tile is constructed by one full brick or diamond and fragments of its neighbours.

The content of a tile is not restricted by the boundaries of the tile as long as the content obeys certain rules. In a way a jigsaw piece could be considered as a rectangular tile with different boundaries for the content.

The 2012 versions of pair traversals were diamond tiles. The oct 2013 version use brick tiles.

The red rectangle below corresponds with the base tile of a thread diagram. The green diamond corresponds with the tile of a pair traversal pattern alias the matrix of a text file, the green square is a distorted version making the matrix clearer. The grey diamonds show other tiles that could be selected from a repeating pattern. The blue arrows show the working orders a lace maker can choose. Flip the undistorted green tile along the X or Y axis, means flip the matrix along a diagonal.

The Flanders template is introduced as a custom diagram. There is also a Flanders alias Torchon rose ground between the [NoInterleaved] patterns: figure 8 of Veronika Irvine's paper

The green grid shows the checkerboard tiles as shown in the paper. The tile in the template has the same size. The red rectangle is a possible diamond tile, it is repeated twice in the tile of the template. The stitches in the red diamond are numbered in working order from A1 to B3, mimicking but not really matching the cells of a matrix.