Goodbye G+

Here lies an archive my public posts on Google+, with topics mainly surrounding mathematics and geometry.



Comments

Post a Comment

Popular posts from this blog

Hyperbolic Hopf Fibrations

Image
Hyperbolic Hopf Fibrations The Hopf Fibration of S^3 is amazing and beautiful.  Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson. nilesjohnson.net/hopf.html To understand it better (and fibrations in general), I recommend this talk by Niles too. www.youtube.com/watch?v=QXDQsmL-8Us It turns out there is an analogue of the Hopf fibration for H^3.  In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case.  There is a 2-parameter family of them, plus one additional fibration that does not fit the family.  As with S^3, fibers in the H^3 cases are geodesics.  They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration.  I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this. www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf I'm posting three early pictures of H^3 fibrations belo...
Read more

76 Unique Honeycombs

76 Unique Honeycombs Last weekend, Tom Ruen and I hit the milestone of uploading to wikipedia at least one image for 9 families of compact, Wythoffian, uniform H3 honeycombs, a total of 76 unique honeycombs. http://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space You can easily browse all the images on my wiki user page: http://commons.wikimedia.org/wiki/User:Roice3 - Compact means the cells are finite in extent.  - Wythoffian means we can generate them using a kaleidoscopic construction, that is by reflecting in mirrors.    - Uniform means they are vertex transitive and have uniform polyhedral cells. There may even be more honeycombs that meet all these criteria, I don't know. ( update: see Tom's comment below! )  I do know there are hundreds more which don't meet one or more of these criteria, many undiscovered.  In fact, there are infinitely more because there are some infinite families of honeycombs.  wendy krieger continues to discover and enumerate more...
Read more

A honeycomb of bucky-balls

Image
A honeycomb of bucky-balls Here are two lovely honeycombs, the bitruncated {5,3,5} and bitruncated {3,5,3}.  The first is a honeycomb of bucky-balls!  (a.k.a. truncated icosahedra, which look like soccer balls.) Both are cell transitive , meaning every cell is exactly the same as every other.  They are also vertex transitive , so every vertex is identical as well.  Since they are vertex transitive, this also means they are examples of uniform honeycombs. I think they are edge transitive too!  A truncated icosahedron is not edge transitive, so how can a honeycomb built of them suddenly be edge transitive?  On a truncated icosahedron, some edges join two hexagons, and some join 1 hexagon and 1 pentagon, so you can't move one edge type to another as a symmetry.  But on the honeycomb, all edges are connected to 2 hexagons and 1 pentagon. These two honeycombs are so close to being regular, but do fail in one respect.  They are not face transitive because each has two kinds of faces. ...
Read more