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.

# 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.

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?

The LEGO system offers possibilities far beyond its core identity as a children’s toy. In particular it allows for near-instantaneous experimentation with mechanisms, while at the same time allowing for sophisticated and beautiful constructions that demonstrate physical and mathematical principles. I will demonstrate this by presenting some old and new inventions and discoveries (as well as a few mysteries) involving linkages, unusual gearing, mechanical computation, recursion and more. I will also discuss practicalities of getting started with the medium as well as some of its limitations.

LEGO® is a trademark of the LEGO Group of companies, which does not sponsor, authorize or endorse this content.

According to Poincaré, "geometry is the art of reasoning well from badly drawn figures" [1895]. In this talk I will give an informal discussion of some famous attempts to draw mathematical figures: some more, and some less, badly drawn. I will finish by discussing some of my own work (with Henry Segerman) in this direction, attempting to use computer graphics, interactive web apps, and 3D printing to illustrate mathematics.

A recording of the talk is available here: https://www.youtube.com/watch?v=iaI6lsqL-_0.

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