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Roice Nelson
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Wish I could go!
Our Brilliant Geometry exhibition is open! Come see it at Summerhall, Edinburgh through June 4th. http://www.maths.ed.ac.uk/~aar/brilliantgeometry/
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5/13/17
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My short talk on visualizing hyperbolic honeycombs at the twelfth Gathering for Gardner, based on joint work with +Henry Segerman. Difficult to fit our paper contents into 10 minutes!

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Poincaré and the Early History of 3-manifolds

Lovely article on manifolds and topology. Very nice to see a variety of core concepts and history in one place. I learned a lot!

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More VR, this time with a less common Thurston geometry. I hope there will be eight papers in this series!
Second paper on non-euclidean virtual reality: H²×E, with Vi Hart, Andrea Hawksley, and Sabetta Matsumoto. https://arxiv.org/abs/1702.04862
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New paper on hyperbolic virtual reality, with Vi Hart, Andrea Hawksley and Sabetta Matsumoto. https://arxiv.org/abs/1702.04004
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An interesting article, on the life and work of Maryam Mirzakhani, an Iranian mathematician and immigrant, now working at Stanford, who is subject to Trump's "Muslim ban" and was the first woman to win the Fields Medal - the so called Nobel Prize for Mathematics


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Inverse Square Law

This video is not new. It is over 10 years old. The mathematics is more than an order of magnitude older (17th century).

I like being enamored with the latest and greatest discoveries, but sometimes think about how we could spend a lifetime trying to understand the simplest formula.

Links

https://en.wikipedia.org/wiki/Inverse-square_law

http://www.gravitation3d.com

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artistic view of the 120-cell

Screen capture circa 2008 from this program:
http://www.gravitation3d.com/magic120cell
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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-skeleton of the honeycomb.

Note how we could fit even more ortho-sticks into this structure. An additional ortho-stick would fit inside each rhombic dodecahedron to connect up the 6 ideal vertices, each line connecting two antipodes.

Here was my approach to making this image. A rhombic dodecahedron is a Catalan solid, i.e. a dual to a uniform solid, so I thought the honeycomb might be dual to one of the uniform honeycombs, which I knew how to draw. Catalan solids share the same symmetry group as their duals, so I went looking in the Wikipedia list (link below) for the right reflection group, and suspected {3,4,4} was it. If so, it would just be a matter of finding the right starting edge(s) in the fundamental simplex to reflect around.

I knew the edges would have one ideal point (only one choice in the simplex for this) and one finite point. I also knew the finite point was at the intersection of these ortho-sticks, which determined it. (The simplex is what is called an orthoscheme, and just one point of the simplex would yield ortho-sticks.) Both points are vertices of the fundamental simplex.

However, the resulting 1-skeleton wasn't quite right. It had too many edges and wasn't a well-formed honeycomb - faces would have been skew. The extra edges are precisely the ortho-sticks I described adding to the interior of rhombic dodecahedra above. An "alternation-like" operation to remove ortho-sticks having an odd number of reflections across one of the mirror planes did the trick.

So this is not a "Catalan" honeycomb as suspected, but is tied up with the symmetry group. It is almost the dual of the rectified {4,4,3}. Thurston was one of the creators of Not Knot, and he also discusses the Borromean rings in his book Three-Dimensional Geometry and Topology. There, he says the hyperbolic structure of the link complement can be built from two ideal octahedra. Well, the simplest honeycomb of the {3,4,4} reflection group is composed of ideal octahedra, further reinforcing that connection.

One last observation. Isn't it strange how some of the ideal vertices appear closer than some of the finite ones?

Relevant Links

Not Knot:
https://youtu.be/AGLPbSMxSUM
https://youtu.be/MKwAS5omW_w

Uniform Paracompact Honeycombs:
https://en.wikipedia.org/wiki/Paracompact_uniform_honeycombs

Catalan Solids:
https://en.wikipedia.org/wiki/Catalan_solid

Orthoscheme:
https://en.wikipedia.org/wiki/Schl%C3%A4fli_orthoscheme

Three-Dimensional Geometry and Topology:
https://www.amazon.com/Three-Dimensional-Geometry-Topology-Vol-1/dp/0691083045/

Rectified {4,4,3}:
https://en.wikipedia.org/wiki/Square_tiling_honeycomb#Rectified_square_tiling_honeycomb

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{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.
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