Talk:Cantic cubic honeycomb

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the quarter-cubic construction

I don't understand how the VF's symmetry group changes when you tilt the CD symbol. —Tamfang (talk) 01:26, 10 June 2012 (UTC)[reply]

You're right, it doesn't without clearer markup, but attemping a symmetry implication of the double-bracketing. It should be done via indices on the nodes showing symmetry correspondence. Tom Ruen (talk) 01:53, 10 June 2012 (UTC)[reply]

p.s. Here's a markup of the correspondences between Euclidean groups. So if indexed nodes have ring equivalence, they can be raised in symmetry from left to right. Coxeter/Johnson notation has <> for corresponding 2 nodes, and extra [] for a folded doubling. [4[]] is used for a complete cyclic equivalence. Tom Ruen (talk) 20:29, 10 June 2012 (UTC)[reply]

My kingdom for a dictionary! No, well, how about my chair?

And what does cantic mean? —Tamfang (talk) 00:39, 18 October 2013 (UTC)[reply]

It has to do with the strange linearized halfing Coxeter diagram , which Coxeter calls h₂{4,3,4}, ringing the 3rd node. So cantic means "Cantellated half".

I had seen this before with the polychora, but hadn't seen a definition. I also updated the operational summary at Uniform_polychoron#Geometric_derivations_for_46_nonprismatic_Wythoffian_uniform_polychora.

Here's an example on George's website for : [1]

( )--4--o----(o)----o : Truncated hexadecachoron [as “cantic tesseract” h2{4,3,3}; not counted, duplicate of 17]

Tom Ruen (talk) 02:07, 18 October 2013 (UTC)[reply]

Norman Johnson, 25 Aug 2004 

My names for uniform figures obtained by Wythoff's construction from regular polychora (4-polytopes) or cellulations (3-honeycombs) {p, q, r} are:

          (o)------o-------o-------o    the original    {p, q, r}
               p       q       r

           o------(o)------o-------o    the rectified       "
               p       q       r

          (o)-----(o)------o-------o    the truncated       "
               p       q       r

          (o)------o------(o)------o    the cantellated     "
               p       q       r

          (o)------o-------o------(o)   the runcinated      "
               p       q       r

           o------(o)------o-------o    the birectified     "
               p       q       r

          (o)-----(o)-----(o)------o    the cantitruncated  "
               p       q       r

          (o)-----(o)------o------(o)   the runcitruncated  "
               p       q       r

          (o)-----(o)-----(o)-----(o)   the omnitruncated   "
               p       q       r

          ( )------o-------o-------o    the half     {p, q, r}  (p even)
               p       q       r

          ( )------o------(o)------o    the cantic       "         "
               p       q       r

          ( )------o-------o------(o)   the runcic       "         "
               p       q       r

          ( )------o------(o)-----(o)   the runcicantic  "         "
               p       q       r

          ( )-----( )------o-------o    the snub     {p, q, 3}  (q even)
               p       q

          ( )-----( )-----( )------o    the snub rectified {p, q, r}
               p       q       r                                (r even)