The graph theory of blackwork embroidery

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Blackwork embroidery, also known as "Spanish stitch'' or "Holbein stitch'', is a needlework technique often associated with Elizabethan England. The patterns used in Blackwork generally are strongly geometric, and are traversed in a way that can be easily described by graph theory. We will characterize these graph traversals and present some algorithms that may be of practical use to needleworkers as well as theoretical interest.

The graph theory of blackwork
embroidery

Joshua Holden
Department of Mathematics
Rose-Hulman Institute of Technology

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Until now, most methods for making a hyperbolic plane from crochet or similar fabrics have fallen into one of two categories. In one type, the work starts from a point or line and expands in a sequence of increasingly long rows, creating a constant negative curvature. In the other, polygonal tiles are created out of a more or less Euclidean fabric and then attached in such a way that the final product approximates a hyperbolic plane on the large scale with an average negative curvature. On the small scale, however, the curvature of the fabric will be closer to zero near the center of the tiles and more negative near the vertices and edges, depending on the amount of stretch in the fabric. The goal of this project is to show how crochet can be used to create polygonal tiles which themselves have constant negative curvature and can therefore be joined into a large region of a hyperbolic plane without significant stretching. Formulas from hyperbolic trigonometry are used to show how, in theory, any regular tiling of the hyperbolic plane can be produced in this way.

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1. The graph theory of blackwork embroidery Joshua Holden Department of Mathematics Rose-Hulman Institute of Technology