Posts

Showing posts from November, 2016

Updated version of my paper with Roice Nelson on the arXiv today.

Image
Originally shared by Henry Segerman

Updated version of my paper with Roice Nelson on the arXiv today. This version to appear in the Journal of Mathematics and the Arts. https://arxiv.org/abs/1511.02851


Bitruncated {4,4,4} Flythrough

Bitruncated {4,4,4} Flythrough

What's the natural next step after spherical images? Spherical animations flying through the honeycombs! This is fun to watch using the YouTube app on your phone, where you can pan by moving your phone around.

http://youtu.be/Y6ZI1kxZoFU

John Baez suggested trying this particular honeycomb, as well as reversing the coloring. I really like the bitruncated honeycombs, especially those derived from self-dual symmetry groups because they end up nearly regular, sometimes even completely regular.

Like the honeycomb of buckyballs I've posted about, this one is cell-transitive, edge-transitive, and vertex-transitive, meaning you can take any of these elements to any other of the same type and preserve the overall honeycomb symmetry. However, faces are nontransitive since there are squares and octagons. Compare with these other bitruncations, which are also transitive for all elements but faces.

Bitruncated {5,3,5}
http://en.wikipedia.org/wiki/Order-5_do…

My brother shared this short illustrated book with me a few years ago and it's gold.

Image
My brother shared this short illustrated book with me a few years ago and it's gold. The world would be a better place if we all read and internalized it.

I also like to visit the wikipedia page listing logical fallacies for a reasoning checkup every now and then. It has grown a bit unwieldy though.

https://en.wikipedia.org/wiki/List_of_fallacies

Fighting the mother of all biases is a constant battle, making it difficult to avoid succumbing to logical failures, but the effort is worth it.

https://en.wikipedia.org/wiki/Confirmation_bias

Originally shared by Sean Walker

Bad Arguments

I got my son the book Bad Arguments recently. He quite enjoyed it and immediately thereafter started helpfully categorizing my perspectives on subjects we discussed in terms of them. =D

In the book, Ali Almossawi categorizes and describes bad logical arguments. Which both helps kids (and adults!) identify when others are stumbling in their logic, but also disciplines their own thinking processes by helpin…

Hyperbolic Honeycomb Photospheres

Originally shared by Roice Nelson

Hyperbolic Honeycomb Photospheres

Encouraged by Tom Ruen again, I fired up the honeycomb rendering code this week. The result is approximately 70 new honeycomb images for Wikipedia and the album below with spherical images of some of my favorites. Have a look around... literally!

This batch focused on uniform paracompact honeycombs. Uniform means they are vertex-transitive and paracompact means some cells have ideal vertices or centers. See if you can pick out some of the cell types.

There are many more paracompact honeycombs to render, but this was a nice chunk, covering all those with Coxeter diagrams that are linear graphs.

Relevant Links

Uniform Paracompact Honeycombs
The new Wikipedia images can be found here:
https://en.wikipedia.org/wiki/Paracompact_uniform_honeycombs

Visualizing Hyperbolic Honeycombs
This paper by Henry Segerman and myself gives an introduction to exotic honeycombs like these. We focus on regular honeycombs, but much of the content…

Hyperbolic Honeycomb Photospheres

Hyperbolic Honeycomb Photospheres

Encouraged by Tom Ruen again, I fired up the honeycomb rendering code this week. The result is approximately 70 new honeycomb images for Wikipedia and the album below with spherical images of some of my favorites. Have a look around... literally!

This batch focused on uniform paracompact honeycombs. Uniform means they are vertex-transitive and paracompact means some cells have ideal vertices or centers. See if you can pick out some of the cell types.

There are many more paracompact honeycombs to render, but this was a nice chunk, covering all those with Coxeter diagrams that are linear graphs.

Relevant Links

Uniform Paracompact Honeycombs
The new Wikipedia images can be found here:
https://en.wikipedia.org/wiki/Paracompact_uniform_honeycombs

Visualizing Hyperbolic Honeycombs
This paper by Henry Segerman and myself gives an introduction to exotic honeycombs like these. We focus on regular honeycombs, but much of the content is relevant.
https://arxiv.org/ab…