Knotology weavers

Pattern for a knotology weaver.

A knotology weaver can weave mixes of tabby weave, corrugated kagome, and voxel weaving. Remarkably, all these weaves are woven with 100% fabric coverage using the same pattern of potential folds (not all potential folds necessarily being used in any given passage.)

The pattern of creases in a knotology weaver (see image above) can be described as a right-angled triangle wave with "altitudes" dropped from each apex to the sides of the strip. These two categories of creases are termed diagonal (solid lines in the image above) and perpendicular (dashed lines.)

Voxel weaving only ever uses the perpendicular creases.

Tabby weave, when done according to the universal method (i.e., with a hole in the center of each square and weavers running at 45°) does not use the perpendicular creases at all; it only uses the diagonal creases when passing over an edge (e.g., one of the 12 edges of a cubical basket.)

Corrugated kagome always uses the perpendicular creases (at fold angles of ±90°) to form its peak or dell caps, and uses the diagonal creases (at varying fold angles) when passing between two triangles of the same salience (i.e., peak-to-peak or dell-to-dell) or when crossing over an edge (e.g., one of the 6 edges of a tetrahedral basket.) When passing between triangles of opposite salience the diagonal crease is not folded.

The squares in voxel weaving have sides equal to the weaver width; the squares woven in universal tabby weave have sides equal to the length of the diagonal creases—larger by a factor of √2. Corrugated kagome must meet voxel weaving essentially at a truncated cube corner. Corrugated kagome must meet universal method tabby weave essentially at an octahedron equator.