Shiying Dong

This family of crocheted surfaces is an extension of the group I displayed at the 2024 Bridges Art Exhibition. It is comprised of five punctured Klein bottles with 0, 1, 2, 3, and 4 boundary components. Each has symmetries that are not obvious in the most common presentation of a Klein bottle in 3-dimensions. The middle surface is a twice-punctured Lawson or Sudanese Klein bottle. The top left surface is closed with two crosscaps. It is made from two annuli bounded by a common Hopf link in orange and blue. The top right has the centerline graph the simple figure 8 graph. The boundary of the bottom right has three components, and each pair wraps around each other three times. The observer is encouraged to discover their symmetry groups.

This piece is in the same series as my exhibit pieces in Bridges 2023 and JMM 2024. In these, I use the topological crochet technique to build a yarn sculpture that manifests the chiral icosahedral symmetry. This surface is nonorientable with genus 86, bounded by a link of six coils that intertwine each other symmetrically. The construction starts from a foundation chain graph made from 12 (2,5)-torus knots lining up like a dodecahedron, with saddle joints appropriately set up, using a method I discovered during topological crochet exploration. This method was taught in a workshop in Bridges 2024. Patterns of all models I present here will be included in the book "Unravelling Topological Crochet," which Eve Torrence and I are working on.

This is a set of two thrice-punctured projective planes. Both centerline graphs are the 1-skeleton of the tetrahedron. The boundary of the left forms a (3,3)-torus link, and on the right, the Borrmean rings. Capping these punctures with disks restores immersions of the projective plane in the 3d space. The left results in the famous Boy’s surface with a 3-fold symmetry, and the right gives rise to an undiscovered immersion with chiral tetrahedral symmetry. This new immersion sheds more light on the intrinsic structure of the projective plane.

In the landscape of polylinks, the link formed by the ten triangles aligned with the ten pairs of antipodal faces in the icosahedron is unique. Such a link bounds this piece. The centerline graph is the 1-skeleton of the truncated icosahedron. This surface is nonorientable with genus 22, with chiral icosahedral symmetry.

This piece is similar to the Borromean rings, as it is formed by three identical structures linked in a way that has chiral tetrahedral symmetry. However, in this piece, each structure is a Solomon’s knot, a link of two components. The centerline graph is the 1-skeleton of the truncated octahedron. This surface is orientable with genus 6, with chiral octahedral symmetry.