Rectified {4,4,3}
Rectified {4,4,3}
I've posted images of 5 rectified hyperbolic honeycombs so far, but there are 15 total, one for each regular honeycomb. I finished up images of the complete set this weekend, but thought it'd be nice to go through the remaining one-a-day (in random order), so they can get some individual attention. If you want to jump straight to seeing all of them, the full set is on my wiki user page: https://commons.wikimedia.org/wiki/User:Roice3
The regular {4,4,3} honeycomb has {4,4} cells and a {4,3} vertex figure. {4,4} denotes a Euclidean square tiling. The {4,3} is a cube.
Therefore, the rectified {4,4,3} has rectified {4,4} cells and cubical cells. A rectified {4,4} tiling is interesting because it is just another {4,4} tiling. Two properties lead to this. First, a rectified square is just another smaller square. Second, the vertex figure of a {4,4} is a square.
See if you can find the two cell types in the images: cubes and square tilings. Unlike many of the other rectified honeycombs, this one has only one kind of face (squares), so I find it especially pretty.
The green circles are an artifact of my finite computer. This includes the green boundary in the fisheye view. The real honeycomb repeats indefinitely, and these areas would appear filled in.
I've posted images of 5 rectified hyperbolic honeycombs so far, but there are 15 total, one for each regular honeycomb. I finished up images of the complete set this weekend, but thought it'd be nice to go through the remaining one-a-day (in random order), so they can get some individual attention. If you want to jump straight to seeing all of them, the full set is on my wiki user page: https://commons.wikimedia.org/wiki/User:Roice3
The regular {4,4,3} honeycomb has {4,4} cells and a {4,3} vertex figure. {4,4} denotes a Euclidean square tiling. The {4,3} is a cube.
Therefore, the rectified {4,4,3} has rectified {4,4} cells and cubical cells. A rectified {4,4} tiling is interesting because it is just another {4,4} tiling. Two properties lead to this. First, a rectified square is just another smaller square. Second, the vertex figure of a {4,4} is a square.
See if you can find the two cell types in the images: cubes and square tilings. Unlike many of the other rectified honeycombs, this one has only one kind of face (squares), so I find it especially pretty.
The green circles are an artifact of my finite computer. This includes the green boundary in the fisheye view. The real honeycomb repeats indefinitely, and these areas would appear filled in.
If you used a horocentric projection (ie the view from an "ideal point", the contents of the squares would alternate with the view through the cube, ie x4o3o, and directly to an adjacent o4x4o. The lads and i on the lists i frequent have been fiddling around with some truly interesting projections, such as Mercartor (infinite equator, finite polar).
ReplyDeletePhantastic !!
ReplyDeleteThat sounds really cool wendy krieger!
ReplyDeleteIs the horocentric projection you describe somewhat like an orthographic projection? The views here are from an ideal point as well, but it is then a perspective view into the ball model. So things don't line up as neatly as you're describing.
I would love to see some of your lists' projections.