The theoretical discovery of hyperbolic geometry first got its actual tactile example in 1868 when Eugenio Beltrami created a negatively curved surface from paper annuli and named it *a pseudosphere*. Later the name *pseudosphere* got attached to a surface created by a tractrix rotating around its axis. However, mathematicians found more useful for theoretical purposes using different, non-tactile models such as Klein or Poincare disc models or half-plane model. Those are traditionally used in college textbooks. However, to experience deeper understanding of hyperbolic geometry, these models were not enough for Bill Thurston when he was a college student. Since in 1901 Hilbert proved that hyperbolic plane cannot be described analytically in 3-space, Thurston together with his peers at informal seminar decided to make a tactile model of hyperbolic plane and created it by gluing together paper annuli without knowing about Beltrami’s paper model created hundred years earlier. I learned about Thurston’s model in 1997 and decided to make it more durable by crocheting it. Crocheted hyperbolic planes have turned out to be a useful tool in tactile explorations of hyperbolic geometry giving to theoretical knowledge a different perspective.

# Seminar

The Illustrating Math Virtual Seminar meets the second Friday of each month. Talks cover a wide range of topics related to successes and challenges of mathematical illustration, from cutting edge theoretical research to explorations of intersections between mathematics and the arts. The seminar showcases innovative ways to communicate and explore deep mathematical ideas.

The monthly seminar is held on Zoom. Each meeting opens with two five-minute ‘show and ask’ style presentations (volunteer here to give one), which are followed by the main feature, a 40 minute invited colloquium talk. Immediately afterwards, participants (and speakers) are invited to gather informally on the illustrating math discord server for further social interaction.

The Zoom room for the seminar is : https://wlu.zoom.us/j/99070727614

I'll tell a few mathematical stories from my personal experience with mathematical illustration as a research tool in number theory, sharing some of my experience with the process, not just results. Topics include Apollonian circle packings, Mobius transformations, continued fractions, and algebraic integers.

I'll tell a few mathematical stories from my personal experience with mathematical illustration as a research tool in number theory, sharing some of my experience with the process, not just results. Topics include Apollonian circle packings, Mobius transformations, continued fractions, and algebraic integers.

When most people picture mathematical art, they envision regular shapes formed by plotting precise equations. In this talk, I will discuss my approach to creating art through a very different process, where mathematics is used to guide a virtual growth process. The results frequently mirror shapes found in nature, both in form and aesthetic appeal.

Nelson Max will show segments of several computer animated films from the 1970s, on turning a circle or sphere inside out by a regular homotopy, and on space filling curves, and demonstrate a web-based system for uniform tilings of the sphere, plane, and hyperbolic plane.

I’ve been illustrating mathematics since I’ve known any, and will be showing a variety of work in a range of media.

I'll talk about my work in mathematical visualization: making accurate, effective, and beautiful pictures and models of mathematical concepts. I'll discuss what it is that makes a visualization compelling, and show many examples in the medium of 3D printing.

PLEASE NOTE THE UNUSUAL TIME OF THIS MONTH'S MEETING!

By combining mathematics, technology, and craft we can produce physical and visual artifacts that help us discover and explore new ideas. We'll discuss a range of topics from knot theory to the fiber arts, and use code, 3D printing, and vintage punch card knitting machines to create pattern generation tools and physical models that illustrate and clarify the underlying mathematics. We'll also discuss the ups and downs of design processes that include challenges that arise from intersecting mathematics, technology, and crafting.

Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object – this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure. At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity. The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. We have developed a formalization of the topology of two-periodic weft knitted textiles using a construction we call the swatch [1]. Using this construction, we can prove that all two-periodic weft knits form ribbon links [2]. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics.

The process of illustrating mathematics is in some ways the reverse of mathematical modelling. Instead of trying to make systems in mathematics that reflect the physical world, it tries to create images and objects in the world and that reflect mathematical structure. What does it mean to tie models as closely as possible to abstract ideas, and how can this help both study mathematical ideas and also find new applications for mathematics, especially in manufacturing?

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