Isonemal Prefabrics with No Axes of Symmetry

TL;DR

This paper refines Roth's taxonomy of isonemal prefabrics with no axes of symmetry, addressing their existence in different sizes, and their behavior under doubling and halving, with implications for woven cube designs.

Contribution

It completes the classification of certain isonemal prefabrics, specifically types 33--39, and analyzes their properties related to size, doubling, halving, and symmetry.

Findings

Identifies which designs exist in various sizes.

Determines which prefabrics remain isonemal after doubling or halving.

Discusses implications for woven cube constructions.

Abstract

This paper refines Richard Roth's taxonomy of isonemal weaving designs through the final types 33--39 in order to complete the solution of three problems for those designs: which designs exist in various sizes, which prefabrics can be doubled and remain isonemal, and which can be halved and remain isonemal. These types have no symmetry axes but have quarter-turn symmetries. Jean Pedersen's problem of woven cubes is also discussed.