Fabric, instead of paper, has been used to fold pleats since ancient Rome and Egypt. Textile pleating is still used today in the fashion industry. Compared to paper fibers, woven cloth fibers are able to move freely or be distorted, making cloth an ideal medium to fold intricate, yet easy-to-produce, tessellations. In fabric origami tessellation, points marked on the cloth are tied and knotted together, allowing the resulting gathering of the fabric to flatten into pleats on the other side. The technique of tying and knotting fabric is also known as smocking, an embroidery technique with its basic honeycomb pattern that was used extensively in the 18th and 19th centuries to provide an elastic quality to non-elastic textiles. Today, smocking is used by designers to create specialized decorative patterns that blend traditional craft with modern aesthetics. One of these modern explorations that are rooted in smocking is fabric origami. In fabric origami, points marked on the cloth are tied and knotted together similar to smocking. However, unlike smocking in which the fabric is left to move freely and wrinkle, fabric origami allows the resulting gathering of the fabric to flatten into pleats on the other side. Origami artist Chris Palmer has used this technique to create a wide array of tessellation designs, which he called “Shadowfolds,” to be used as decorative textiles.
Fabric Origami is a series of ongoing works that are based on a technique the artist calls “tessellation grafting.” To graft a tessellation, one starts by cutting along all the edges figuratively, creating a new tessellation by inserting rectangles, again, figuratively, along all the edges and polygons connecting the vertices. If a vertex in the original tessellation has a valence of three, then the polygon connecting the now-separated vertices must be a triangle. If a vertex in the original tessellation has a valence of four, then the polygon connecting the new vertices must be a quadrangle. In general, for vertices that are n-valent, the inserting polygons must be n-gons. To make the fabric origami, the corners of new polygons are sewn together, collapsing the polygons back to points and rectangles back to lines. On one side of the sewn fabric, the seams are now reflecting the original pattern. On the other side of the fabric, the fabric is carefully pleated and flattened to create a new tessellation. Interestingly, the tessellation grafting is rooted in the dual graph of a plane graph G in graph theory.
Link to research on grafting tessellation for fabric origami