Inspired by William Thurston's description of the complement of the Borromean rings (Section 3.4, "The Geometry and Topology of Three-Manifolds"), this pair of quilts demonstrates his construction and is especially useful in visualizing how to deform the spanning 2-complex---shown from above in Quilt I and from below in Quilt II (when open)---into the boundary of an octahedron. Using the quilts, we perform this deformation by pulling on the cording and gathering each arc to a point (a vertex). In doing so, each quilt closes up into an octahedron, with 8 fabric faces, 12 stick edges, and 6 cording vertices. The colorings on the faces, edges, and vertices indicate how to identify the octahedra to form the complement of the Borromean rings. The video below assumes some familiarity with the Thurston/Menasco method for representing the complement of a knot or link as two ideal polyhedra with face pairings. For a clear and well-illustrated explanation of this general method, see Chapter 2 of Jessica S. Purcell's Hyperbolic Knot Theory. |