Textile D forms and $D_{4d}$

TL;DR

This work bridges the mathematical theory of D-forms with textile practice by presenting textile realizations such as pita-form accessories and biscornu. It employs embroidery-based symmetry visualization, notably using $D_{4d}$, to illustrate how complex 3D shapes emerge from paired developable surfaces and seams. The paper contributes concrete designs (a pita-form handbag and an Orpheus hat) and a symmetry sampler consisting of eleven biscornu that realize $D_{4d}$ and its subgroups via hitomezashi embroidery, highlighting multiple subgroups including $S_8$, $D_4$, $C_{4v}$, $C_4$, $D_2$, $C_{2v}$, and $C_s$. The findings offer educational and creative value, providing tangible objects to teach three-dimensional geometry, symmetry, and their connections to molecular and algebraic concepts.

Abstract

D-forms have in the past been created from inflexible materials, or considered as abstract mathematical objects. This paper describes a number of realisations of D-forms, and the related pita-forms, in textiles. Examples are given in which the created 3D space is purposefully employed and others in which ornamentation of the constituent surfaces is the highlighted feature. In particular, a set of biscornu has been fashioned to provide a 3D sampler of the axial point group $D_{4d}$ and its subgroups, using hitomezashi.

Figures (19)

• Figure 1: A crocheted stadium, executed in plarn (yarn made from plastic bags) by the author in 2015.
• Figure 2: The non-convex from which a trisphericon can be obtained as a pita-form. This figure also appears in KS.
• Figure 3: The flat form devised by Atkinson for a pita-form handbag circa 2016. Courtesy Bruce Atkinson.
• Figure 4: The pita-form bag itself; is is large enough to hold an A4 notepad. Courtesy Bruce Atkinson.
• Figure 5: Orpheus hat, knitted and photographed by author, pattern by Norah Gaughan for Berrocco NG.