Flight in Curved Spaces

I was recently reminiscing about the movie Not Knot, and began thinking about recreating some of the hyperbolic scenes. But then today I stumbled upon a really nice application that does exactly what I was thinking about and more:

http://www.geometrygames.org/CurvedSpaces/index.html

Its a pretty small download, works on most platforms, very fun.

More polytopal animations

I was inspired by a still image from the Wikipedia entry on the 24-cell to make an animation of rotating and then stereographically projecting the 24-cell, showing the images of the original polytope edges. Math on Wikipedia is pretty spotty, but the polytope entries have really become top-notch in the last few years.
Here's a still from the animation:



and a medium and small resolution version of the animation:

small (300x300, 6 MB)
medium (512x512, 17 MB).

600-cell animations

Over the last year or so, when I have time, I've been trying to fine tune ray-traced animations of polytopes (using Sage, Tachyon, ffmpeg, and cddlib). My latest effort was to do a loop of Schlegel projections of the beautiful 600-cell. The 600-cell is one of the six regular polytopes in 4 dimensions; its usually considered the 4D analog of the icosahedron. Here's a still from my animation:



Since Blogger's re-encoding butchers my movies, I will just post links. Here are four sizes:

small (320x200, 3.5 MB)
medium (640x400, 7 MB)
large (1280x800, 13 MB)
large, high-res (1280x800, 33 MB).

Twenty Four Cell, take 2

Hmm...blogger re-encodes my video and for some reason their version looks like crap, so I won't even leave it up. I think my new attempt has much better lighting, and the faces have translucent panes now. But here is a still shot:



Three sizes of the new version:
small (320x200)
medium (640x400)
large (1280x800).

24 cell movie

This is basically a test of my workflow for a more ambitious project, but its somewhat amusing in its own right. The original 48 MB version is here (much better quality).

Polytopes in Sage, take 1

Here's a first attempt at a short video intro to polytopes in Sage. Sage 4.0, which should be released soon, has some new functionality that was added during Sage Days 15 (thanks to David Perkinson for reviewing some of that, and Rob Beezer for helping David learn the review ropes).

The video is about 4 MB and can be found here. I am not sure how the format will work on other computers so if anyone takes a look and it doesn't work let me know. Also, I am interested in getting constructive criticism on the video.

Permutohedron mirrors

The image below is from a viewpoint in a mirror-faced 3D permutohedron (truncated cuboctahedron). It should link to a larger (1920x1200) version. This image was produced with Sage (using the Tachyon raytracer and some new code for polytopes that I've been working on).

Slices of the 600-cell

Over the last 6 months or so I've been doing some work on visualizing polytopes, Groebner fans, and other geometric/algebraic objects. I gave a presentation last week about some of that, which forced me to finish up some projects. One of those was a movie of the 600-cell being sliced by 3-planes.

Schlegel and the 600 cell

I just wrote a patch to do Schlegel diagrams (a sort of projection) of 4D polytopes in Sage. The 4D regular polytopes are an awful lot of fun to think about; below is a picture of the 600 cell as rendered by my new code. Its much more fun to play with it interactively - check it out.