![Very Happy :D](./images/smilies/grin.gif)
![Smile :)](./images/smilies/smile.gif)
I had trouble figuring out how to typeset the lace city in ASCII, so I decided to let graphviz to do it for me instead:
![Image](http://hddb.teamikaria.com/dl/3Q0XMZ2ASJN5RCA5Y3JAA54HMD.png)
(In retrospect this is probably quite easy to do in ASCII, but it wasn't obvious just from looking at the coordinates!
![Glaring :glare:](./images/smilies/glare.gif)
![Sweatdrop :sweatdrop:](./images/smilies/sweatdrop.gif)
![Rolling Eyes :roll:](./images/smilies/rolleyes.gif)
![Glaring :glare:](./images/smilies/glare.gif)
![Embarassed :oops:](./images/smilies/redface.gif)
As is obviously, this CRF has trigonal symmetry... What may not be immediately obvious, though, is that it also has trigonal symmetry in perp space, so it actually has 3,3-duoprism symmetry.
![Smile :)](./images/smilies/smile.gif)
There are 18 icosahedra in 3 groups of 6, each group of which has icosahedra sharing faces with the connectivity of a trigonal prism.
![Surprised :o](./images/smilies/shock.gif)
![Surprised :o](./images/smilies/shock.gif)
Here's a parallel projection centered on a hexagon shared by two J92's:
![Image](http://hddb.teamikaria.com/dl/JFVGYWAGY76MCX63F52ERRRE8M.png)
The green and yellow cells are J92's, in case it's not obvious (it's not easy to get good renders on this beast). The blue and purple cells are regular icosahedra -- there are 3 pairs visible here. Between the pairs, are clusters of 5 square pyramids, that look like a fragment of the rectified 600-cell o5o3x3o; within each pair, however, there are 5-fold clusters of tetrahedra that look like a fragment of the 600-cell. You can see some of the tetrahedron-trigonal prism-tetrahedron combos that seem to be a recurring theme in these CRFs.
This structure looks deceptively simple... until you look at what goes on in the perpendicular ring to the six J92's:
![Image](http://hddb.teamikaria.com/dl/JDM2GRSCQFAKYAWJFPWXY4HYFK.png)
The J92's are outlined in red here. As you can see, there are 6 icosahedra here, with the 3 in top sharing only vertices with the 3 at the bottom. These are actually two different trigonal-prism clusters of icosahedra; the top and bottom clusters of pentagon-like faces of the projection are images of 6 other icosahedra that share faces with the top 3 icosahedra and the bottom 3 icosahedra, respectively.
Before we get to that, though, let's look at something else: the trigonal prisms in this 4D viewpoint:
![Image](http://hddb.teamikaria.com/dl/D80FRJ2JWPAZ8PXGFA3ARW1HQE.png)
As you can see, there are 4 triangular prisms visible here, each of which connects to two tetrahedra (the ones touching the center shown in blue). Now, notice that the pentagonal cross-sections of the icosahedra between the outer (red) triangular prisms form a 5-3-5-3 configuration around the vertices of the central triangular prism. Sounds familiar?
![Smile :)](./images/smilies/smile.gif)
![Laughing :lol:](./images/smilies/lol.gif)
Well, let's look at a slightly shifted viewpoint that shows a cluster of 6 icosahedra in trigonal prism arrangement:
![Image](http://hddb.teamikaria.com/dl/NAXBGWPWAZWATHMKW1QJENBG83.png)
Notice how similar this is to the snub 24-cell? Yet it's not quite the same, because it has trigonal symmetry instead of tetragonal!
![Surprised :o](./images/smilies/shock.gif)
Alright, let's finish off with another view of the J92's, this time around a shared triangular ridge:
![Image](http://hddb.teamikaria.com/dl/RFHZYP30F0S2TV05ZA48YT4B4N.png)
This time, we show the J62 (metabidiminished icosahedra) wedges between the J92 cells instead. The left side of the projection seems to have the outline of a cube, but that's an illusion. (There is actually no edge between the two "square faces" of the "cube".)