Talk:Architectonic and catoptric tessellation

I haven't looked how names correspond but these cell-transitive (uniform dual honeycombs) articles exist: Tom Ruen (talk) 20:50, 30 March 2013 (UTC)

• Disphenoid tetrahedral honeycomb - dual of Bitruncated_cubic_honeycomb
• Rhombic dodecahedral honeycomb - dual of Tetrahedral-octahedral honeycomb
• trapezo-rhombic dodecahedral honeycomb dual of Gyrated tetrahedral-octahedral honeycomb

And referenced names of uncreated articles

• Hexakis cubic honeycomb dual of Truncated cubic honeycomb
• Trigonal trapezohedral honeycomb dual of Quarter_cubic_honeycomb
• Square bipyramidal honeycomb dual of Rectified_cubic_honeycomb
• Prismatics:
  • Tetrakis square prismatic tiling dual of Truncated_square_prismatic_honeycomb
  • Cairo pentagonal prismatic honeycomb dual of Truncated_square_prismatic_honeycomb
  • Triakis triangular prismatic honeycomb dual of Truncated hexagonal prismatic honeycomb
  • Rhombille prismatic honeycomb dual of Gyrated tetrahedral-octahedral honeycomb
  • Deltoidal trihexagonal prismatic honeycomb dual of Rhombitrihexagonal_prismatic_honeycomb
  • Kisrhombille prismatic honeycomb dual of Truncated_trihexagonal_prismatic_honeycomb
  • Floret pentagonal prismatic honeycomb dual of Snub hexagonal prismatic honeycomb

It seems the names given by Conway are:

• Obtetrahedrille corresponds to Disphenoid tetrahedral honeycomb
• Dodecahedrille corresponds to Rhombic dodecahedral honeycomb
• no name given to Trapezo-rhombic dodecahedral honeycomb
• Pyramidille corresponds to Hexakis cubic honeycomb
• Obcubille corresponds to Trigonal trapezohedral honeycomb
• Oboctahedrille corresponds to Square bipyramidal honeycomb
• no names given to the duals of the prismatic uniform honeycombs

-- Toshio Yamaguchi 22:20, 31 March 2013 (UTC)

Looks consistent. I'll have to look at my copy later. Tom Ruen (talk) 00:17, 1 April 2013 (UTC)

I am not sure whether the image for the cell of the Obcubille is correct. In the illustration on page 294 this looks more like a rhombohedron than a trigonal trapezohedron. -- Toshio Yamaguchi 08:38, 1 April 2013 (UTC)

I copied information from the uniform honeycombs to help. It's funny Conway used nicely labeled flat vertex figure diagrams for the 4D polychora, but hard-to-see 3D vertex figures for the honeycombs. Tom Ruen (talk) 21:06, 1 April 2013 (UTC)

Looks awesome so far, thank you. I don't know whether I can do this before next weekend, but I will try to make exploded views of the vertex figures if the program I intend to use for this permits it as I think this gives a better impression of the structure of those honeycombs. I guess that there are more vertex configurations than those listed in the book. For example, the oboctahedrille has two types of vertices, one with 6 cells around the vertex and one with 12 cells around the vertex. -- Toshio Yamaguchi 21:28, 1 April 2013 (UTC)

I'm glad for any new graphics. I might be more interested in Conway's names if he had also expanded them in the related finite 4D uniform polychorons and their duals, but he just has his ambo notation (related to Wythoff construction active mirrors) there for the uniform polychora and nothing on duals (chapter 26, p.392-403). At least Johnson's names for the uniform polychora are in the index. Tom Ruen (talk) 22:09, 1 April 2013 (UTC)

I finshed adding the cells for the architectonic tessellations. Perhaps this article should be renamed Architectonic and Catoptric tessellation given a focus on this subset of uniform honeycombs, and the Catoptrics are hard to define without their duals. Tom Ruen (talk) 00:56, 8 April 2013 (UTC)

Thanks. I agree with this renaming as the table lists corresponding ones together and you are right that the Catoptrics really are the result of the existence of their duals. -- Toshio Yamaguchi 06:45, 8 April 2013 (UTC)

words

Rather than "Elongated pyramid", which also means something else, how about "isosceles pyramid"? —Tamfang (talk) 04:47, 2 March 2014 (UTC)

Agreed. Tom Ruen (talk) 05:34, 2 March 2014 (UTC)

why only the cubish?

The text needs to say why this list excludes fifteen of the convex uniform tessellations of E3. The term catoptric is derived from a Greek word for mirror, so I can see excluding those that cannot be generated by mirrors (without alternation), but I think there are only three of those. —Tamfang (talk) 21:38, 16 March 2018 (UTC)

I added a statement. I see the prismatic forms were excluded from Archimedean solids since they are an infinite family there, while that's not true for the stacked forms here, so an argument could be made for inclusion. It looks like we don't have the prismatic dual honeycombs at all. Are you interested in rendering them? Tom Ruen (talk) 00:46, 17 March 2018 (UTC)

I should have known better than to look at this page when I'm not going to have any free time before bed! —Tamfang (talk) 05:20, 17 March 2018 (UTC)