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# The 17 Worlds of Planar Ants

### Canada/USA Mathcamp, July 26 and 27, 2021.

#### Online, by invitation only.

Abstract. My goal is to get you hooked, captured and unreleased until you find all 17 in real life, around you.

We all know that the plane can be filled in different periodic manners: floor tiles are often square but sometimes hexagonal, bricks are often laid in an interlaced pattern, fabrics often carry interesting patterns. A little less known is that there are precisely 17 symmetry patterns for tiling the plane; not one more, not one less. It is even less known how easy these 17 are to identify in the patterns around you, how fun it is, how common some are, and how rare some others seem to be.

Gotta Catch 'Em All!

Handout: 17Worlds.pdf

Annotated Slides: 17Worlds@.pdf

 Links. g  r  b  Jeff2207

# Knots in Three and Four Dimensions

### Canada/USA Mathcamp, July 27, 2021.

#### Online, by invitation only.

Abstract. Much as we can understand 3-dimensional objects by staring at their pictures and x-ray images and slices in 2-dimensions, so can we understand 4-dimensional objects by staring at their pictures and x-ray images and slices in 3-dimensions, capitalizing on the fact that we understand 3-dimensions pretty well. So we will spend some time staring at and understanding various 2-dimensional views of a 3-dimensional elephant, and then even more simply, various 2-dimensional views of some 3-dimensional knots. This achieved, we'll take the leap and visualize some 4-dimensional knots by their various traces in 3-dimensional space, and if we'll still have time, we'll prove that these knots are really knotted.

Slides: K34-Slides.pdf

Annotated Slides: K34-Slides@.pdf