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Showing posts from January, 2017

Borromean Rings Complement

Borromean Rings Complement

I've watched the famous Not Knot video about complements of knots and links many times over the years. My dad actually brought it home one day when I was in high school in the mid-90s, at which point it seemed so "out there" that a friend and I found it comical. Given that beginning, it's funny how much I've come to connect with it.

I revisited it again this weekend. I figured attempting to draw the video's finale image myself would lead to fun insights, and the exercise did not disappoint. So here is a pannable image of what Not Knot calls the "hyperbolic structure of the link complement" for the Borromean rings.

It's a honeycomb of rhombic dodecahedra, each with 6 ideal vertices and 8 finite vertices. If you stare out it for a bit, you will notice ortho-sticks, i.e. a set of 3 mutually orthogonal axes. These triplets of infinitely long lines connect up with each other at the ideal points to build up the entire 1-s…

{4,3,∞} Domains

{4,3,∞} Domains

This is one way to picture the simplex domains of the {4,3,∞} reflection group. To understand the meaning of {4,3,∞}, see the paper Visualizing Hyperbolic Honeycombs.

https://arxiv.org/abs/1511.02851

The domains are colored light/dark based on their depth mod 2 in domain adjacency graph (also described in the paper).

If the domains were drawn filled, we would instead see coloring on the surface of a ball, but I trimmed them by growing spheres that cut into each simplex from its 4 vertices.
https://youtu.be/vUBNXVfZlkA

Knotted Surfaces

Knotted Surfaces

I enjoyed watching this series just now (about an hour spread over 4 videos). Great use of dimensional analogy, and I finally have a nice mental picture of a knotted sphere.

Originally shared by Arnaud Chéritat

Ester Dalvit just released on YouTube a series of videos about knots in 4D. A realization of high quality and excellent explanations.
https://www.youtube.com/playlist?list=PLyxHTRWELFBr9TP8jx400bPGP-ECsBufz