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...at the January 2009 Joint Mathematics Meetings in Washington, DC. It was held Wednesday, January 7, 2009, 1:00 p.m.-5:50 p.m. and was organized by sarah-marie belcastro and Carolyn Yackel. (Information on the 2005 Special Session is here.) The Special Session and Exhibit were written up in the ICMS News.
The schedule was as follows. Speaker names link to their webpages, and speaker titles link to their abstracts (plain text). Photos, where available, are linked below as well; click on any image to enlarge it.
3:00--3:30 Mathematical Fiber Arts Exhibit (juried) photos (and detailed descriptions of exhibit items)
3:30--3:50 Mary Shepherd (Southwest Missouri State): Visualizing Groups and Subgroups in Counted Cross Stitch.
4:00--4:20 Daina Taimina (Cornell
University): Exploring two-dimensional manifolds with
4:30--4:50 Joshua Holden (Rose-Hulman
IT): Braids, Cables, and Cells: An intersection of Mathematics,
Computer Science, and Fiber Arts.
Amy F. Szczepanski
Calculating Patterns for Knitted Surfaces.
We present a method for calculating the pattern for knitting some geometric shapes (such as spheres and tori) that can be described as surfaces of revolution. Each of these shapes can be knit as a series of circular rounds. Writing the pattern can be reduced to a problem of determining how many stitches should be in each round and how many rounds are needed. Some approximations will need to be done, as rounds must have a whole number of stitches, and the overall pattern must have a whole number of rounds. The number of stitches in each round can be calculated by using a parameterization of the curve, rotation matrices, and approximations of arclength. We present a description of the method and software that has been written to calculate some patterns.
Temari Math and Geometry on the Sphere.
We will discuss the interplay between mathematics and the craft of embroidered temari thread balls. Two different ways of classifying temari ball designs will be considered; one presented by Conway, Burgiel, and Goodman-Strauss in their recent book The Symmetries of Things and the other as a pro jection of polyhedra onto the sphere. One portion of the talk will demonstrate the use of temari as examples of spherical symmetries or polyhedral pro jections. The other half will focus on which symmetries or polyhedra can be realized in some mathematically exact sense given the restriction to temari techniques.
Exploring two-dimensional manifolds with crochet hook.
Crocheted models of the hyperbolic plane are very inviting to play with. There are many different ways to fold symmetric hyperbolic planes. Starting from the same basic plane, there are unlimited possibilities to create different fiber sculptures. Do these different shapes have anything to do with mathematics or are they just purely aesthetic forms? All those surfaces are geometric 2-manifolds. Each of them is covered by the hyperbolic plane and so each is locally isometric to the hyperbolic plane (and to each other). However, among these geometric 2-manifolds, only the hyperbolic plane is simply connected—all the other hyperbolic surfaces have holes or circles that cannot be shrunk on the surface. In the first /Fiber Arts in Mathematics and Mathematics Education/ session I showed crocheted hyperbolic octagon that forms a two-manifold. In this talk I will show how to make two-manifolds from crocheted hyperbolic rectangular hexagons and ideal triangles.
Diaper Pattern in Needlepoint.
Informally stated, a diaper pattern in decorative art is one that has visual diagonals in two different directions. Needlepoint canvas, because of its evenweave construction, is well suited to the creation of diaper patterns. The relationship of diaper patterns to the 17 wallpaper groups will be discussed. Many examples of needlepoint diaper patterns will be shown, including how the use of color in a single pattern can aid in identification of the symmetry.
A semiregular tessellation of a plane is a tessellation using only regular n-gons such that each vertex meets an adjacent shape in a vertex, and the configurations are the same at all vertices (up to rotation). There are only finitely many semiregular tessellations (eight or nine, in addition to the three regular tessellations, depending on how you count). I will discuss how to render semiregular tessellations in quilt form, in particular, what are the geometric shortcuts that save time but possibly not the fabric.
Braids, Cables, and Cells: An intersection of Mathematics, Computer Science, and Fiber Arts.
The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including knitting and crochet, where braids are called “cables”. We will view some examples of braids and their mathematical representations in these media.
Visualizing Groups and Subgroups in Counted Cross Stitch.
Symmetry groups have a rich algebraic structure, and since many are non-Abelian, they comprise a nice set of examples in which to explore general ideas about groups, free from the hidden pitfalls that students sometimes encounter in Abelian groups. Using counted cross stitch examples created by the author, this talk will address: (1) visualizing the individual group elements and the group operation, (2) visualizing subgroups and cosets in both finite and infinite groups, and (3) exploring other properties and theorems related to groups and subgroups. We will begin these explorations looking at the finite group D4 and the infinite wallpaper group p4m.
Braid words in generalized helix stripe patterns
When multiple strands of yarn are used in knitting (as, for example, when knitting with more than one color), a knitter twists a pair of strands when switching from using one to using another. The yarn strands between the knitting and the balls of yarn get tangled as the knitting proceeds. In turn, this produces a braid in the strands. This talk will consider the special case of the braid words generated by generalized helix stripe patterns. The basic technique produces spiralling stripes of row-height one. (Standard striping produces cylindrical or line-segment stripes.) We will first explain how to generalize helix striping to thicker stripes and different numbers of colors. We will then determine which braid words are generated using generalized helix stripe patterns. (This has practical applications in terms of detangling yarn while knitting.)
Don’t Blow a Gasket!
The beautiful Sierpinski Gasket (also called the Sierpinski Triangle) can be created in many different ways. In this talk, which grew out of the speaker’s experience in tatting a Sierpinski triangle, we’ll look at a few of those ways and how they naturally map into the fiber arts.