Talk:Isohedral figure

message: Face-transitive is a standard term for polytopes. Face-uniform was an incorrect term derived from uniform polytopes which are are vertex-transitive, and their duals which are face-transitive.
* Yes - Tom Ruen 22:27, 3 February 2007 (UTC)
replies: text: I wasn't sure whether it was best to move the pages first, before doing all that. Yes I know it can be both with redirects, but we still need to decide the titles of the pages to which we redirect the others. Steelpillow 20:09, 6 February 2007 (UTC)
thread_title: Poll to move to face-transitive

message: I see r-hedral apparently means r shapes of tiles, monohedral (r=1), dihedral (r=2), trihedral (r=3)...  So maybe this could be defined somewhere else, and then monohedral could be moved there. Monohedral tiling directs to Tessellation#Introduction_to_tessellations. Tom Ruen (talk) 19:09, 19 August 2015 (UTC)
replies: text:  We don't usually use redirect pages for discussions, but how about moving this to talk:Monohedral tiling? I could undelete that page, which contained just the maths rating template. — Sebastian 19:18, 19 August 2015 (UTC)
replies: text:  It wouldn't be good to have talk page on a redirect. Monohedral figure directs here at the moment, so I'll just add a new section for now. Tom Ruen (talk) 19:24, 19 August 2015 (UTC)
thread_title: Monohedral figures

message: The lede says: "Isohedron redirects here", but it doesn't.  This article actually links to Isohedron!  (If that statement were correct, this link would create a circular – and useless – self-reference.)
Also, Pentagonal tiling#Reinhardt (1918) links to an old name for this page, Isohedral, which redirects to this article's page, i.e. to Isohedral figure.  (According to earlier talk on this discussion page, the expression "Isohedral figure" was a compromise between alternatives such as (IIRC) "Isohedral polytope" and "Isohedral polygon", whilst trying to avoid continuing to use the adjective "Isohedral" for a page title.)
Another difficulty for readers at present is that the discussion of Isohedral figures and Isohedral is disjointed, being split across both of the named pages.  Readers would get a clearer picture from a single, well-organised page that combines information from both.
Here's a plan:
#Remove the erroneous redirection statement I quoted above.
#Merge this page, Isohedral figure, with Isohedron.
#Choose a sensible name for the resulting page, one which meets users' expectations that page names are the names of things, i.e. nouns, and also one which reflects the broader context, rather than:
:* one restricted context (e.g. "Isohedral polygon"), or
:* one particular detail (e.g. "Isohedron").
Based on those constraints, I suggest that the best title we could use would be Isohedral polytope.
I look forward to a fruitful discussion with interested readers and editors.  yoyo (talk) 14:50, 5 November 2015 (UTC)
replies: text: There could be some needed corrections, but I think "isohedral figure" is more compehensive than isohedron (or isohedral tilings), since it applies to higher dimensional polytopes and honeycombs. Also I don't think "Isohedral polygon" makes sense as a polygon has no faces. Tom Ruen (talk) 16:29, 5 November 2015 (UTC)

text: We have series of articles on isotoxal figure, isogonal figure, etc. Unless someone want to overturn that, I think that merging the isohedron in here is the best approach. — Cheers, Steelpillow (Talk) 17:37, 5 November 2015 (UTC)

text: Isohedron should be merged with this article because all it has are examples of isohedra. The examples on isohedron should be moved to the examples on this page. Eli355 (talk) 14:46, 7 June 2018 (UTC)
replies: text: The merge has been performed. Eli355 (talk) 00:40, 27 June 2018 (UTC)
thread_title: Factual correction and proposal for a merge with [[Isohedron]]

message: Counterexamples might be worth mentioning, if only on this talk page.
The following are not isohedral figures. They have all identical (congruent) faces, but they are not "face-transitive":
*Rhombic icosahedron (dual of a modified Johnson solid) It has 20 faces that are [golden] rhombi, but is not face-transitive.
*Bilinski dodecahedron (dual of cuboctahedron) It has 12 faces that are [golden] rhombi, like the rhombic dodecahedron (a Catalan solid (all of which are face-transitive)), but is not face-transitive.
These lead to some questions:
*Is the superset: polyhedra that have identical faces (with no other requirement) significant, and does it have a name?
*Is there a larger superset if reflection of asymmetrical faces is also allowed?
*Is there a name for the difference set (polyhedra that have identical faces but are not isohedral)?
Addition?
*The excavated dodecahedron (which is not currently listed as isohedral) has 60 equilateral triangle faces. It is like the pentakis dodecahedron (which is listed), except that its 12 pyramids extend inward rather than outward.
Subtraction??
*The rhombic triacontahedron (which is listed as isohedral), to me, looks different from all the rest. It has two kinds of faces, as does the rhombic icosahedron (which is not isohedral). Mathworld includes it too. I don't understand how it is isohedral. (It's probably me.) - A876 (talk) 23:57, 3 April 2019 (UTC)
thread_title: Counterexamples