Talk:Cantic cubic honeycomb

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)


 * 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)


 * 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)
 * [[File:Coxeter diagram affine rank4 correspondence.png]]

My kingdom for a dictionary! No, well, how about my chair?
And what does cantic mean? —Tamfang (talk) 00:39, 18 October 2013 (UTC)

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.

Here's an example on George's website for :
 * --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)

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)