Rectified {6,3,3}
Rectified {6,3,3}
The regular {6,3,3} is a honeycomb with hexagonal tilings as cells. The vertex figure is a tetrahedron.
http://en.wikipedia.org/wiki/Hexagonal_tiling_honeycomb
When we rectify the {6,3,3}, the hexagonal tilings turn into a uniform tiling with triangles and hexagons called the trihexagonal tiling.
http://en.wikipedia.org/wiki/Trihexagonal_tiling
And we get tetrahedral cells where the vertices used to live.
When I look at these images, my mind tends to see an intricate lattice of tetrahedra connected up at their tips. The vertices are all identical, which is a property of uniform honeycombs (they are vertex transitive). Take a look at a vertex of the rectified {6,3,3} and see if you can deduce the shape of the vertex figure. I'll put the answer in the comments tomorrow.
The regular {6,3,3} is a honeycomb with hexagonal tilings as cells. The vertex figure is a tetrahedron.
http://en.wikipedia.org/wiki/Hexagonal_tiling_honeycomb
When we rectify the {6,3,3}, the hexagonal tilings turn into a uniform tiling with triangles and hexagons called the trihexagonal tiling.
http://en.wikipedia.org/wiki/Trihexagonal_tiling
And we get tetrahedral cells where the vertices used to live.
When I look at these images, my mind tends to see an intricate lattice of tetrahedra connected up at their tips. The vertices are all identical, which is a property of uniform honeycombs (they are vertex transitive). Take a look at a vertex of the rectified {6,3,3} and see if you can deduce the shape of the vertex figure. I'll put the answer in the comments tomorrow.
The vertex figure is variously written as o6$3o3o or xx3oo&#h. The o6x3o3o is a WME (wythoff mirror edge) polytope, its vertex figure, like all WME figures, is a lace prism, usually written by using multiple vertex nodes.
ReplyDeleteBecause only one node is marked, the vertex figure is also a WME, and vertex transitive.
BTW, the designation of this style of presentation is a "spherated edge-frame" it is a view from the edge of a poincare projection.
wendy krieger, which particular "lace prism"? :)
ReplyDeleteI looked up "lace prism" on your polygloss, and you wrote there that it is a generalization of antiprisms. Do uniform prisms falls into this category as well?
Thanks a bunch for the name of this style of presention! I didn't know that term.
It is xx3oo&#h. It looks frightening, but its actually two wythoff polytopes x.3o. and .x3.o laced together with edges of h. In short, it's a triangle prism, of height 1.7320508. The notation is a direct representation of the coordinates, that you can feed in a symbol and get the shape.
ReplyDeleteThe first one is that we can directly read the wythoff symbol and pull the vertex figure.
cool, yep! I could tell you knew exactly what it was, but "triangle prism" was what I wanted to write here since I think more individuals will recognize that vs. the notation (which I should be familiar with myself but am not).
ReplyDeleteAnd your response answered my other question too, that uniform prisms are an example of a lace prism. Thanks again!
I left it for a day too., for the same reason. But i usually calculate most of the stuff like size, directly from the vertix figure. So it's got a specific height too. But lace prisms can handle fostrums, cupolae, antiprisms, pyramids, and lots of other things too.
ReplyDeleteWhen it has two bases, it can be seen as transitive on the lacing. It's kind of like looking at the image of a ladder in a kaleidoscope. The images of the ladder make all of the lacing edges. I'm hoping for the gadget to turn up today or tomorrow to draw piccies.