Talk:Thorold Gosset

list of his regular and semiregular figures

Good start for new article!

I have a list of his regular and semiregular figures at:

User:Tomruen/semiregular_polytopes

I think I pretty much understand what they all are, but could use more pictures. The "semichecks" are sort of weird mixture of tilings and polytopes. Tom Ruen 21:37, 17 April 2007 (UTC)

Yes, I found that page of yours; quite useful for matching Gosset's names to the more modern ones. Someday it would be nice to have a page on semiregular polytopes, although it is slightly problematic since the definition of "semiregular" is not widely agreed upon. The term is probably more of historical interest than anything. Also nice would be a page on Gosset-Elte polytopes, those which Coxeter denotes by k_ij. -- Fropuff 22:41, 17 April 2007 (UTC)

Gosset-Elte polytopes?

I'm curious - what's the complete list of Gosset-Elte polytopes. Maybe all the one-end-ringed single-bifurcating groups? Tom Ruen 23:40, 21 April 2007 (UTC)

I moved my test tables from here to User:Tomruen/Gosset-Elte figures. Tom Ruen 02:29, 24 April 2007 (UTC)

That's all of them except for the A-family given by 0_ij (a type A diagram with any single node ringed). Also I think the B-family is more commonly labeled the D-family (Cartan's notation for the Dynkin diagrams having won out over Coxeter's). The polytopes then fall into something like an ADE classification. Basically, the condition for k_ij to exist is that

${\frac {1}{i+1}}+{\frac {1}{j+1}}+{\frac {1}{k+1}}>1$

or equality if you want to include the honeycombs. -- Fropuff 03:21, 22 April 2007 (UTC)

Very interesting, thanks! I'll have to play with some ij tables for each k. I guess there's also hyperbolic honeycombs on the lesser than side?! I've not seen any lists for bifurcating hyperbolic groups, not dared to guess before.

On group names, it is frustrating to have references from different systems, but not had the heart to convert group names in tables I made at Coxeter-Dynkin diagram. Tom Ruen 06:27, 22 April 2007 (UTC)

Oh yes, for honeycomb groups, I couldn't figure out how to make the ~ hat symbols, etc - Unicodes seemed not to exist in general! Tom Ruen 06:30, 22 April 2007 (UTC)

I was silly on Bimonster, made some Coxeter graphs, but I have no idea what they really are, if there's any geometric reality. Tom Ruen 06:45, 22 April 2007 (UTC)

I ordered a reprint copy of Elte's book, The semiregular polytopes of the hyperspaces.[1] so perhaps can get an article of Gosset-Elte polytope going. Tom Ruen 18:51, 26 April 2007 (UTC)

Nice, I hadn't realized the book was still in print (I guess I never looked). Coxeter mentions that Elte apparently never realized that his polytopes could be realized as members of a single family, so I'm not sure how complete his list is. I suspect he doesn't discuss the A or D family except for their regular/semiregular members. Coxeter probably deserves credit for realizing the entire family. I'm not sure who first called these Gosset-Elte polytopes or how wide spread the name is. I've seen it in print a few times. Though for lack of a better name I suppose it will do. -- Fropuff 19:35, 26 April 2007 (UTC)

Norman Johnson (mathematician) calls them Gosset-Elte figures, perhaps to allow infinite/honeycomb forms, assuming polytope means finite?

Email archive quote: "...The other polytopes k_ij were investigated by E. L. Elte in a 1912 paper.I call the whole set the Gosset-Elte figures." [2]

Tom Ruen 22:50, 26 April 2007 (UTC)

Relatives?

Is Thorold Gosset related to William Sealy Gosset? Michael Hardy 21:24, 20 April 2007 (UTC)

Not that I know of. For what it's worth, there is a program called GOSSET, the documentation of which offers this comment:

“The program is named after the amateur mathematician Thorold Gosset (1869­–1962), who was one of the first to study geometrical structures in six, seven and eight dimensions, and his contemporary, the statistician William Sealy Gosset (1876–­1937), who was one of the first to use statistical methods in the planning and interpretation of agricultural experiments. Although from our geometric viewpoint their work is related, we do not know if the paths of Thorold (Cambridge, London, lawyer) and William Sealy (Oxford, Dublin, brewer) ever crossed.”

-- Fropuff 00:03, 21 April 2007 (UTC)

Nonconvex?

I wonder if he counted some nonconvex ones like sidtaxhi, dattady and gidtaxhi. http://polytope.net/hedrondude/sishi.htm says he did ("...they have been known since Gosset's time..."), but I'm not sure.

Sidtaxhi is small ditrigonal hexacosihecatonicosachoron (small ditrigonal 600-120-cell), its verf is sidtid. Cells are 120 gikes and 600 tets.

Dattady is ditrigonal hecatonicosahecatonicosachoron (ditrigonal dis-120-cell), its verf is ditdid. Cells are 120 gikes and 120 ikes.

Gidtaxhi is great ditrigonal hexacosihecatonicosachoron (great ditrigonal 600-120-cell), its verf is gidtid. Cells are 600 tets and 120 ikes.

I think these three semiregular polychora deserve their own articles, btw, but that's beside the point. 4 T C 05:35, 8 August 2011 (UTC)