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-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}. Update! After a post on Twitter over a year later Tom Ruen quickly figured out this is indeed a Catalan honeycomb, dual to the alternated {4,4,3}, aka dh{4,4,3}. We found it interesting that taking the dual of r{4,4,3} then alternating gave the same result as alternating {4,4,3} then taking the dual. IOW, hdr{4,4,3} = dh{4,4,3} See https://en.wikipedia.org/wiki/Square_tiling_honeycomb#Alternated_square_tiling_honeycomb
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
https://goo.gl/photos/Nqhanf5JBQSvuQqy5
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}. Update! After a post on Twitter over a year later Tom Ruen quickly figured out this is indeed a Catalan honeycomb, dual to the alternated {4,4,3}, aka dh{4,4,3}. We found it interesting that taking the dual of r{4,4,3} then alternating gave the same result as alternating {4,4,3} then taking the dual. IOW, hdr{4,4,3} = dh{4,4,3} See https://en.wikipedia.org/wiki/Square_tiling_honeycomb#Alternated_square_tiling_honeycomb
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
https://goo.gl/photos/Nqhanf5JBQSvuQqy5
This might be suitable mood music for looking at your creation. It's live at the moment but will be available as a recording later. Glass Night https://goo.gl/LnnlbJ
ReplyDeleteIt does seem apropos! And I really like Philip Glass.
ReplyDeleteVisually quite interesting - thank you. But more importantly, I'm learning a lot from the links you provided. For example, I had no idea what the term dual meant in the context of polyhedrons.
ReplyDelete