Talk:Gosset–Elte figures

existence condition[edit]

I get $\sum {\frac {p_{i}}{2{p_{i}}+2}}<1$ (equal for Euclidean tilings). Funny that our different formulae give the same answers.

MAYBE your summation works on more than 3 branches, while simple version doesn't? (indeed, since 1/3+1/3+1/3+1/3!=1! for 4-branch group!) Tom Ruen (talk) 02:41, 10 June 2010 (UTC)[reply]

The only such hyperbolic systems are (6,2,1), (4,3,1), (3,2,2), (2,1,1,1), (1,1,1,1,1). Each Wythoff simplex has a vertex at infinity; the last has five out of six. (I disallow those with a vertex beyond infinity.) —Tamfang (talk) 20:59, 9 June 2010 (UTC)[reply]

Cool! Do you have printed sources for any of this. I have been slowly enumerating the euclidean honeycombs, but no sources on the hyperbolic bifurcating families.

Coxeter lists the linear families in The beauty of Geometry, essay 10: Regular hyperbolic honeycombs in hyperbolic space (2-5 dimensions). (They are on wikipedia at List_of_regular_polytopes#Tessellations_of_hyperbolic_4-space)

Tom Ruen (talk) 21:44, 9 June 2010 (UTC)[reply]

OR, sorry. Some of it may be in Regular Polytopes but I can't find my copies. —Tamfang (talk) 00:52, 10 June 2010 (UTC)[reply]

No, I'm wrong, do have a source, Reflection Groups and Coxeter Groups by James Humphreys has them listed in section 6.9, googlebooks, but that section isn't online. [1]. He differentiates between compact (no vertices or facet centers at infinity?) and noncompact, lists noncompact up to n=10 (none higher). The compact groups are listed at Coxeter-Dynkin_diagram#Hyperbolic_infinite_Coxeter_groups. Tom Ruen (talk) 21:53, 9 June 2010 (UTC)[reply]

Incidentally, my cursory search suggests there are no convex compact systems above H4; do you know if that's right? —Tamfang (talk) 17:58, 10 June 2010 (UTC)[reply]

Yes, all listed above, none above n=5 (H4). Tom Ruen (talk) 22:55, 10 June 2010 (UTC)[reply]

Er, yeah. Sometimes I don't see what's under my nose. —Tamfang (talk) 03:01, 11 June 2010 (UTC)[reply]

This illustration will show how I arrived at that formula. In this table, each row is the unit vector normal to one of the mirrors in 3^3,2,1. I start at the ends of the branches and work inward.

1	0	0	0	0	0	0
-1/2	√3/2	0	0	0	0	0
0	-1/√3	√(2/3)	0	0	0	0
0	0	0	1	0	0	0
0	0	0	-1/2	√3/2	0	0
0	0	0	0	0	1	0
0	0	-√(3/8)	0	-1/√3	-1/2	1/√24

If the system were flat or hyperbolic, the last entry would be zero or imaginary. You can see that each nonzero (and nonbold) entry in the last row depends on the length of a branch; I worked out the first few such numbers before noticing the pattern (which is easily verified by induction). —Tamfang (talk) 04:09, 10 June 2010 (UTC)[reply]

I wondered where I wrote this … —Tamfang (talk) 02:49, 8 June 2023 (UTC)[reply]

Forgot the signs (which are irrelevant to the point I was after, but necessary to the schema). Corrected now. —Tamfang (talk) 17:52, 10 June 2010 (UTC)[reply]

Cool! You're having fun! But wouldn't normal vectors of 3₂₁ be 7-dimensional? Tom Ruen (talk) 17:56, 10 June 2010 (UTC)[reply]

Let me guess: you wrote that question after I inserted the signs and before I added the space between "|" and "-1/2" in the second row. —Tamfang (talk) 17:59, 10 June 2010 (UTC)[reply]

Whoa, yes, I thought I was crazy for a moment!

I'd be having more and different fun if I knew an efficient way to generate a figure in S3 – with, say, 533 symmetry – from a single repeat-unit. —Tamfang (talk) 21:32, 10 June 2010 (UTC) [reply]

Keep trying! I'm completely behind in this level of generation. I ought to be using your 321 hyperplane normal vectors to generate the E7 family polytopes, and then the Petrie polygon projection graphs. That's actually my weakest family for graphics - User:Rocchini made the good graphs. But in general I've had the regular polytopes, and cheated and constructed the truncations/rectifications as geometric/topological operations. Working from the fundamental simplex domain with facet reflections would be much more general, although I'd still have to compute the equal-edge point for all active mirrors. No time except for little distractive puttering now. Tom Ruen (talk) 22:50, 10 June 2010 (UTC)[reply]

Rather than computing that point, you could put the vertex at the origin and the mirrors at a fixed (or zero) distance from it. —Tamfang (talk) 03:01, 11 June 2010 (UTC)[reply]

"all rectified (single-ring) Coxeter–Dynkin diagrams"[edit]

The article has this sentence (in the "Definition" section:) "The simplex family can be seen as a limiting case with k=0, and all rectified (single-ring) Coxeter–Dynkin diagrams."

The second clause seems to be a sentence fragment. What does it mean? --kundor (talk) 16:42, 21 January 2015 (UTC)[reply]

It means that the hexatope {3,3,3,3}, for example, can be seen as 0₄₀, its first rectification as 0₃₁, its birectification as 0₂₂ – each with its ring on the terminal node of a branch of length zero. Took me a minute or two to parse it. If that sentence is still there, I'll look at improving it. —Tamfang (talk) 04:31, 15 June 2023 (UTC)[reply]